Math Problems Vault

Problem 1298 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Examine the concavity or convexity of the curve

y=1+\sqrt[3]{x}

at the points $A(-1,0), B(1,2)$ and $C(0,0)$ .

?

Problem 1299 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=3x^{2}-x^{3} .

?

Problem 1300 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=\frac{a^{3}}{a^{2}+x^{2}} \ (a>0) .

?

Problem 1301 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x+x^{\frac{5}{3}} .

?

Problem 1302 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=\sqrt{1+x^{2}} .

?

Problem 1303 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x+\sin x .

?

Problem 1304 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=e^{-x^{2}} .

?

Problem 1305 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=\ln (1+x^{2}) .

?

Problem 1306 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x\sin (\ln x) \ (x>0) .

?

Problem 1307 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x^{x} \ (x>0) .

?

Problem 1308 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Prove that the curve

y=\frac{x+1}{x^{2}+1}

has three inflection points lying on the same straight line. Sketch the graph of this function.

?

Problem 1309 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

How should the parameter $h$ be chosen so that the "probability curve"

y=\frac{h}{\sqrt{\pi }}e^{-h^{2}x^{2}} \ (h>0)

has inflection points $x=\pm \sigma$ ?

?

Problem 1310 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Examine the concavity or convexity of the cycloid

x=a(t-\sin t), y=a(1-\cos t) \ (a>0)

?

Problem 1311 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Let the function $f(x)$ be twice differentiable on the interval $a \leqslant x<+\infty$ , and:

$f(a)=A>0$ ;
$f'(a)<0$ ;
when $x>a$ , $f''(x) \leqslant 0$ . Prove: the equation $f(x)=0$ has one and only one real root in the interval $(a,+\infty )$ .

?

Problem 1313 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.8 "Concavity and Convexity. Inflection Points"

Prove that the functions

x^{n} \ (n>1), e^{x}, x\ln x

are concave on the interval $(0,+\infty )$ ; while the functions

x^{n} \ (0<n<1), \ln x .

are convex on the interval $(0,+\infty )$ .

?

Problem 1114 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.6 "Higher-Order Derivatives"

Show that the curve $y = \ln x$ is convex upward.

?

Remark.

?

The derivative of the first derivative, i.e., $(y')'$ or $(f'(x))'$ , is called the second derivative and is denoted by $y''$ or $f''(x)$ . The derivative of the second derivative is called the third derivative, and so on. To avoid confusion with an exponent, the order of the derivative is denoted by a number placed in parentheses. Roman numerals are also sometimes used. If at a given point of the curve $y = f(x)$ the quantity $y'' > 0$ , then the curve is convex downward, i.e., toward the negative direction of the $Oy$ axis. If $y'' < 0$ , then in the neighborhood of the given point the curve is convex upward. If $y''$ changes sign as $x$ passes through a given value, then the point on the curve at that $x$ is an inflection point.

Problem 1115 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.6 "Higher-Order Derivatives"

Show that the curve $y = a^{x}$ is convex downward.

?

Problem 1116 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.6 "Higher-Order Derivatives"

Show that the sinusoid $y = \sin x$ is convex upward where $y > 0$ , and downward where $y < 0$ .

?

Problem 1117 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.6 "Higher-Order Derivatives"

Show that the tangent curve $y = \operatorname {tg} x$ is convex downward where $y > 0$ , and upward where $y < 0$ .

?

Problem 1118 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.6 "Higher-Order Derivatives"

Investigate the direction of convexity of the curve $y = ax^{3}+bx$ for $a > 0$ .

?

Problem 1119 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.6 "Higher-Order Derivatives"

Find the inflection points of the curve $y = e^{-x^{2}}$ .

?

Problem 1120 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.6 "Higher-Order Derivatives"

The same question for the curve $y = \frac{1}{1+x^{2}}$ .

?

Problem 2005 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.3 "Convexity, Curvature, and Radius of Curvature"

For what values of $x$ is the curve $y=x^{3}+ax+b$ convex upward?

?

Problem 2006 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.3 "Convexity, Curvature, and Radius of Curvature"

Determine the direction of convexity of the curve $x(x^{2}-y^{2})+y^{2}=0$ at the points where $x=\dfrac {3}{2}$ .

?

Problem 2007 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.3 "Convexity, Curvature, and Radius of Curvature"

Investigate the nature of the concavity of the curve $x^{4}=y(x^{2}-y^{2})$ at points where $y<0$ .

?

Problem 2008 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.3 "Convexity, Curvature, and Radius of Curvature"

Investigate the nature of the convexity of the curve $r\cos^{3}3\varphi = 1$ for $-\dfrac {\pi }{6}<\varphi <\dfrac {\pi }{6}$ .

?

Problem 1.20.48 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Find the intervals of convexity of the function:

?

(1)

$f = x^{\alpha }, \alpha > 1, x > 0$

(2)

$f = x^{\alpha }, 0 < \alpha < 1, x > 0$

(3)

$f = e^{x}$

(4)

$f = \ln x$

(5)

$f = x \ln x$

(6)

$f = \operatorname {arctg} x$ .

Problem 1.20.49 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Find the intervals of convexity and the points of inflection of the function.

?

(1)

$f = 2 x^{4}-3 x^{2}+x-1$ ;

(2)

$f = x^{5}-10 x^{2}+3 x$ ;

(3)

$f = \frac{1}{1-x^{2}}$ ;

(4)

$f = \frac{x^{3}}{12+x^{2}}$

(5)

$f = \sqrt[3]{x+3}$

(6)

$f = \frac{\sqrt{x}}{x+1}$

(7)

$f = \sqrt[3]{4 x^{3}-12 x}$

(8)

$f = x / \sqrt[3]{x^{2}-1}$ .

Problem 1.20.50 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Find the intervals of convexity and the points of inflection of the function.

?

(1)

$f = \cos x$ ;

(2)

$f = x+\sin x$

(3)

$f = e^{-x^{2}}$

(4)

$f = e^{1 / x}$

(5)

$f = \frac{10}{x} \ln \frac{x}{10}$

(6)

$f = x \sin \ln x$

(7)

$f = \operatorname {arctg} \frac{1}{x}$ ;

(8)

$f = e^{\operatorname {arctg} x}$

Problem 1.20.51 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)(3)(4)

Find the points of inflection of the function.

?

(1)

$f = x^{4}-6 x^{2}+5 x$ ;

(2)

$f = x^{4}-12 x^{3}+48 x^{2}$ ;

(3)

$f = 2 x^{4}+2 x^{3}+3 x^{2}+3 x+1$

(4)

$f = \left(x^{2}-1\right)^{3}$

Problem 1.20.52 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Find the points of inflection of the function.

?

(1)

$f = 4 x^{2}+1 / x$ ;

(2)

$f = x^{2} /(x-1)^{3}$ ;

(3)

$f = \sqrt[3]{x^{2}}-\sqrt[3]{x^{2}-4}$ ;

(4)

$f = \sqrt[3]{(1-x)(x-2)^{2}}$ ;

(5)

$f = \left|x-1\right| / x^{2}$ .

Problem 1.20.53 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Find the points of inflection of the function.

?

(1)

$f = \left(x^{2}+1\right) e^{x}$ ;

(2)

$f = x^{3} e^{-4 x}$ ;

(3)

$f = x^{2} \ln x$ ;

(4)

$f = \ln x / \sqrt{x}$ ;

(5)

$f = e^{\cos x}$ .

Problem 1.20.54 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Find the points of inflection of the graph of the function.

?

(1)

$f = 36 x(x-1)^{3}$ ;

(2)

$f = x+36 x^{2}-2 x^{3}-x^{4}$

(3)

$f = 1+x^{2}-x^{4} / 2$ ;

(4)

$f = x^{5} / 20-x^{4}+8 x^{3}-32 x^{2}$ ;

(5)

$f = \left(2 x^{2}-x-4\right) /\left(x^{2}-4 x+4\right)$ ;

(6)

$f = x^{4} /(x+1)^{3}$ .

Problem 1.20.55 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)(3)(4)

Find the points of inflection of the graph of the function.

?

(1)

$f = \sqrt[3]{1-x^{3}}$ ;

(2)

$f = \sqrt[3]{x}-\sqrt[3]{x+1}$ ;

(3)

$f = 5+\sqrt[3]{(x-5)^{5}}$ ;

(4)

$f = \sqrt{\left(8-x^{3}\right) /(3 x)}$ .

Problem 1.20.56 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)(3)(4)

Find the points of inflection of the graph of the function.

?

(1)

$f = e^{2 x-x^{2}}$ ;

(2)

$f = x e^{-(x / 2)^{2}}$ ;

(3)

$f = 2 x^{2}+\ln x$ ;

(4)

$f = e^{-2 x} \sin^{2} x$ .

Problem 1.20.57 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)

Find the points of inflection of the function $f$ :

?

(1)

$f = \frac{1}{\sigma \sqrt{2 \pi }} e^{-x^{2} / 2 \sigma^{2}}, \sigma > 0$ ;

(2)

$f = \frac{a x}{x^{2}+b^{2}}, a \neq 0, b \neq 0$ .

Problem 1.20.58 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)

Investigate the polynomials for points of inflection:

?

(1)

$P_{3}(x) = a x^{3}+b x^{2}+c x+d, a \neq 0$

(2)

$P_{4}(x) = a x^{4}+b x^{3}+c x^{2}+d x+e, a \neq 0$ .

Problem 1.20.59 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

For what values of the parameter $a$ does the function $f = e^{x}+a x^{3}$ have points of inflection?

?

Problem 1.20.60 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Prove that the graph of the function $y(x) = (x+1) /\left(x^{2}+1\right)$ has three points of inflection lying on one line.

?

Problem 1.20.61 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Prove that the points of inflection of the graph of the function $y(x) = x \sin x$ lie on the curve $y^{2}\left(4+x^{2}\right) = 4 x^{2}$ .

?

Problem 1.20.62 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)(3)

Let $f(x) = \begin{cases} x^{3}\left(2+\cos \left(1 / x^{2}\right)\right), & x \neq 0, \\ 0, & x = 0. \end{cases}$

Prove that:

?

(1)

the graph of the function $f(x)$ has a tangent at the point $(0 ; 0)$ ;

(2)

the graph of the function $f(x)$ passes from one side of the tangent to it at the point $(0,0)$ to the other side;

(3)

the point $(0 ; 0)$ is not a point of inflection of the graph of the function $f(x)$ .

Problem 1.20.63 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)

?

(1)

Can a point of inflection of a function be a point of extremum of it?

(2)

Can a function that is everywhere convex downward (upward) have more than one extremum?

Problem 1.20.64 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)

Prove that for any twice differentiable function:

?

(1)

between two points of extremum there lies at least one point of inflection;

(2)

between points of inflection of a function there may be no points of extremum at all.

Problem 1.20.65 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)

Prove that:

?

(1)

every polynomial of odd degree $n > 1$ has at least one point of inflection;

(2)

every polynomial of even degree with positive coefficients has no points of inflection.

Problem 1.20.66 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

Let $f(x)$ be a continuously differentiable function on the interval $(a ; b)$ , and suppose that for any points $x_{1}, x_{2}$ of this interval there exists a unique point $c$ such that

\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}} = f^{\prime }(c)

Prove that $f(x)$ has no points of inflection.

?

Problem 1.20.67 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.20.2 "Problems"

(1)(2)(3)(4)

Find the points of inflection of the graph of the function $y = f(x)$ given parametrically by the equations:

?

(1)

$x = t e^{t}, y = t e^{-t}, t > 0$

(2)

$x = \frac{t^{2}}{t-1}, y = \frac{t^{3}}{t-1}, t > 2 ;$

(3)

$x = \frac{2 t^{2}+2}{t}, y = \frac{t^{3}+3 t+1}{t^{2}}, 0 < t < 1$ ;

(4)

$x = \frac{t^{2}-2 t-5}{t^{2}+10 t+25}, y = \frac{t^{2}-4 t+5}{t^{2}+4 t-5}, t > 1$ .

Problem 1.23.5 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

(1)(2)(3)

Let the function $f$ be defined and twice continuously differentiable on $\mathbb {R}$ , and let $f^{\prime \prime }$ not change sign on $\mathbb {R}$ . Prove that:

?

(1)

the equation $f(x) = 0$ cannot have more than two real roots;

(2)

if $f\left(x_{0}\right) f^{\prime }\left(x_{0}\right) < 0, f\left(x_{0}\right) f^{\prime \prime }\left(x_{0}\right) < 0$ , then the equation $f(x) = 0$ has a unique root in the interval $(x_{0} ; x_{0}-f\left(x_{0}\right) / f^{\prime }\left(x_{0}\right))$ ;

(3)

if $f^{\prime }\left(x_{0}\right) = 0, f\left(x_{0}\right) f^{\prime \prime }\left(x_{0}\right) < 0$ , then the equation $f(x) = 0$ has one root each in the intervals $\left(-\infty ; x_{0}\right),\left(x_{0} ;+\infty \right)$ .

Problem 1.7.237 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.7.2 "Problems"

(1)(2)(3)(4)

A function $f$ is called concave (convex) on an interval $X$ if for any $x_{1}, x_{2} \in X$ and any $\alpha \in [0 ; 1]$ the inequality holds

f\left(\alpha x_{1}+(1-\alpha ) x_{2}\right) \geqslant \alpha f\left(x_{1}\right)+(1-\alpha ) f\left(x_{2}\right)

(respectively $f\left(\alpha x_{1}+(1-\alpha ) x_{2}\right) \leqslant \alpha f\left(x_{1}\right)+(1-\alpha ) f\left(x_{2}\right)$ ).

The graph of a function concave on a closed interval $[a ; b]$ lies not below the line drawn through the points $(a ; f(a)),(b ; f(b))$ .

Prove that the function:

?

(1)

$y = a x^{2}+b x+c$ is convex on $\mathbb {R}$ for $a > 0$ and concave on $\mathbb {R}$ for $a < 0$ ;

(2)

$y = a^{x}$ is convex on $\mathbb {R}$ ;

(3)

$y = \log_{a} x$ is concave on $(0 ;+\infty )$ for $a > 1$ and convex on $(0 ;+\infty )$ for $0 < a < 1$ ;

(4)

$y = \sin x$ is concave on $[0 ; \pi ]$ and convex on $[-\pi ; 0]$ .

Problem 1.7.238 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.7.2 "Problems"

Indicate the intervals of concavity and convexity of the function:

?

(1)

$y = \left|x\right|$ ;

(2)

$y = x^{3}$ ;

(3)

$y = \frac{1}{x}$ ;

(4)

$y = \frac{2 x-5}{x+1}$

(5)

$y = \operatorname {ch} x$ ;

(6)

$y = \operatorname {sh} x$ ;

(7)

$y = \lg \left|x\right|$

(8)

$y = \left|\ln x\right|$ .

Problem 1.7.241 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.7.2 "Problems"

(1)(2)(3)(4)

Indicate the intervals of concavity and convexity of the function:

?

(1)

$y = \operatorname {tg} x, x \in (-\pi / 2 ; \pi / 2)$

(2)

$y = \cos^{2} x, x \in (0 ; 2 \pi )$ ;

(3)

$y = \arcsin x, x \in [-1 ; 1]$ ;

(4)

$y = \operatorname {arcctg} x, x \in \mathbb {R}$ .

Concavity and inflection points of graphs

References