Investigation of Functions

$f = 4 x^{3}-21 x^{2}+18 x+7$;

$f = 8 x^{3}-x^{4}$;

$f = x^{5}-5 x^{4}+5 x^{3}-1$;

$f = (x-1)^{3}(2 x+3)^{2}$.

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

$f = e^{x} / x$;

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

$f = x^{\alpha } e^{-x}, x > 0, \alpha > 0$;

$f = x^{2}-10 \ln x$;

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

$f = x / \ln x$;

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

$f = \operatorname {arctg} x-\ln x$;

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

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

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

$f = \frac{x^{3}}{3-x^{2}}$;

$f = \frac{(3-x)^{3}}{(x-2)^{2}}$.

$f = \sqrt[3]{x} /(x+50)$;

$f = \sqrt{8 x^{2}-x^{4}}$;

$f = \sqrt{2 x^{3}+9 x^{2}}$;

$f = x \sqrt{(x+1)^{3}}$;

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

$f = (x+1) \sqrt{x^{2}-1}$;

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

$f = \frac{\sqrt{1+\left|x+2\right|}}{1+\left|x\right|}$;

$f = \frac{\sin x+\cos x}{1+\left|\cos x\right|}$.

$x^{2} y^{2}+y = 1, y > 0$;

$x^{3} y^{3} = x-y, x > 0$.

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

$f(x) = \frac{a^{2}-1}{3} x^{3}+(a-1) x^{2}+2 x$

$f(x) = a x-\sin x$;

$f(x) = a x+3 \sin x+4 \cos x$;

$f(x) = (8 a-7) x-a \sin 6 x-\sin 5 x$

$f(x) = 4 x+\frac{a+1}{2} \ln \left(x^{2}-x+1\right)+\frac{a-5}{\sqrt{3}} \operatorname {arctg} \frac{2 x-1}{\sqrt{3}}$.

if a function $f(x)$ is increasing at every point of some interval, then $f(x)$ is increasing on this interval;

the function $f(x) = \begin{cases} x+x^{2} \sin (2 / x), & x \neq 0, \\ 0, & x = 0, \end{cases}$ is increasing at the point $x = 0$, but is not increasing on any interval containing this point.

$y = x^{3}-4 x^{2}$

$y = x(x-3)^{2}(x+1)^{3}$

$y = 2 \sin x+\cos 2 x$

$y = \frac{1}{x^{3}-2 x^{2}-x+2}$

$y = (x-5) e^{x}$

$y = x^{2} e^{1 / x}$

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

$y = \frac{\sqrt[3]{1+\left|x\right|}}{1+\left|4 x+5\right|}$.

$y = x^{4}-8 x^{2}+12$;

$y = \begin{cases} x^{2}+1, & x \neq 0 \\ 2, & x = 0\end{cases}$

$y = \frac{x^{4}}{4}-2 x^{3}+\frac{11}{2} x^{2}-6 x+3$;

$y = x^{4}-4 x^{3}+6 x^{2}-4 x+5$;

$y = \left(x^{3}-10\right)(x+5)^{2}$;

$y = (x+2)^{2}(x-3)^{3}$.

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

$y = \frac{x}{x^{2}+4}$;

$y = \frac{(x-1)^{2}}{x+1}$;

$y = \frac{(2-x)^{3}}{(3-x)^{2}}$;

$y = \frac{x^{3}+2 x^{2}}{(x-1)^{2}}$

$y = \frac{x^{4}}{(x+1)^{3}}$.

$y = \sin x+\frac{1}{2} \sin 2 x$;

$y = \sin^{3} x+\cos^{3} x$;

$y = \frac{\sin x}{2+\cos x}$;

$y = x+\sin x$;

$y = x-2 \sin^{2} x$;

$y = x-2 \operatorname {arctg} x$;

$y = (x-2) \cos \pi x-\frac{1}{\pi } \sin \pi x$;

$y = \left(x^{2}+1\right) \operatorname {arctg} x-\frac{\pi }{4} x^{2}-x$.

$y = (x-1) e^{3 x}$;

$y = \left(3-x^{2}\right) e^{x}$;

$y = \left(x^{2}-8\right) e^{-x}$;

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

$y = x^{4} e^{-x^{2}}$;

$y = (x+1)^{5} e^{-x}$

$y = \left(x^{3}+3 x^{2}+6 x+6\right) e^{-x}$;

$y = (x+2) e^{1 / x}$.

$y = \sqrt{x} \ln x$;

$y = \frac{\ln^{2} x}{x}$;

$y = \frac{1}{\ln \left(x^{4}+4 x^{3}+30\right)}$

$y = x^{2}-4 x-1-\ln \left(x^{2}-4 x+4\right)$;

$y = \ln \cos x-\cos x$;

$y = \ln \left(x^{2}-1\right)-2 \operatorname {arctg} x$.

$y = x+\sqrt{3-x}$;

$y = \sqrt[3]{x^{3}-2 x^{2}+x}$;

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

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

$y = \sqrt[3]{x^{2}}-\sqrt[3]{x^{2}-1}$;

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

$y = \sqrt[3]{(x-2)^{2}(x-4)^{2}}$;

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

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

$y = 1-\sqrt[3]{(x+1)^{2} /(x+2)^{2}}$.

$y = x^{x}$;

$y = x^{1 / x}$.

$y = \left|x-5\right|(x-3)^{3}$;

$y = \max \left\{ 7 x-6 x^{2}, \left|x^{3}\right|\right\}$;

$y = \sqrt{1+2\left|x-1\right|} /(6+\left|3 x-2\right|)$;

$y = \sqrt[3]{x^{2}\left|2-x\right|}$;

$y = \left|x-1\right| \sqrt[3]{x+2}$;

$y = \left|x^{2}-1\right| e^{\left|x\right|}$;

$y = \left|x^{2}-4\right| e^{-\left|x\right|}$;

$y = e^{-\left|x-1\right|} /(x+1)$.

$y = \sin (x+1)-\left|\cos x\right|, x \in (0 ; \pi )$;

$y = \sin \left|x-3\right|+\cos x, x \in (0 ; \pi )$;

$y = \frac{1+\left|\cos x\right|}{2+\cos x+\sqrt{3} \sin x}, x \in (0 ; \pi )$.

$y(x) = \begin{cases} x, & x \leqslant 0, \\ x \ln x, & x > 0 ;\end{cases} \quad$

$y(x) = \begin{cases} 1+x, & x \leqslant 0, \\ (x)^{x^{3}}, & x > 0 ;\end{cases}$

$y(x) = \begin{cases} 1+x, & x \leqslant 0 \\ x^{x^{2} \ln x}, & x > 0\end{cases}$

$y = (x+1)^{n} e^{-x}, n \in \mathbb {N}$;

$y = \left(1+x+\frac{x^{2}}{2 !}+\ldots +\frac{x^{n}}{n !}\right) e^{-x}, n \in \mathbb {N}$;

$y = x^{k}(1-x)^{n}, k, n \in \mathbb {N}$;

$y = a e^{p x}+b e^{-p x}$, $a, b, p \in \mathbb {R}$.

$x^{3}+y^{3} = 3 x^{2}$;

$x+y = x y(y-x),\left|y\right| < \left|x\right|$;

$x^{4}-y^{4} = x^{2}-2 y^{2}, y > \left|x\right|$;

$\left(x^{2}-y^{2}\right)(x-y) = 1, y > \left|x\right|$.

$x = \frac{1}{t(t+1)}, y = \frac{(t+1)^{2}}{t}, t > 0$;

$x = \ln \sin \frac{t}{2}, y = \ln \sin t$.

$f^{\prime }(0)$ does not exist, $g^{(n)}(0) = 0, n \in \mathbb {N}$;

$f(x)$ and $g(x)$ have a strict minimum at the point $x = 0$;

$f(x)$ and $g(x)$ are not decreasing on any interval $(-\delta ; 0), \delta > 0$, and are not increasing on any interval $(0 ; \delta ), \delta > 0$.

$f^{(n)}(0) = g^{(n)}(0) = 0$

$f(x)$ has a strict minimum at the point $x = 0$; $g(x)$ has no extremum at the point $x = 0$.

$y = x^{3}-6 x^{2}+9, x \in [-1; 2]$;

$y = \frac{1}{2} x^{3}-9 x^{2}+48 x, x \in [0 ; 9]$

$y = 2 x^{3}+3 x^{2}-120 x+100, x \in (-4 ; 5]$;

$y = x^{4}-8 x^{2}+3, x \in [-1 ; 2]$;

$y = x^{5}-5 x^{4}+5 x^{3}+1, x \in [-1 ; 2]$.

$y = \frac{9}{x}+\frac{25}{1-x}, x \in (0 ; 1)$;

$y = \frac{1-x+x^{2}}{1+x-x^{2}}, x \in [0 ; 1]$;

$y = \frac{x^{2}+1}{x^{2}+x+1}, x \in \mathbb {R}$;

$y = \frac{x^{4}+1}{x^{2}+1}, x \in [-1 ; 1]$.

$y = x-2 \sqrt{x}, x \in [0 ; 5]$;

$y = x-2 \ln x, x \in [3 / 2 ; e]$;

$y = x \ln (x / 5), x \in [1 ; 5]$;

$y = \left|x^{2}+2 x-3\right|+1,5 \cdot \ln x, x \in [1 / 2 ; 2]$;

$y = (x-3) e^{\left|x+1\right|}, x \in [-2 ; 4]$;

$y = x^{x}, x \in (0 ; 1]$.

$y = 2 \sin x+\sin 2 x, x \in [0 ; 3 \pi / 2]$;

$y = \cos^{2} x+\cos^{2}(\pi / 3+x)-\cos x \cos (\pi / 3+x), x \in \mathbb {R}$;

$y = 4 x+9 \pi^{2} / x+\sin x, x \in [\pi ; 2 \pi ]$

$y = 2 \operatorname {arctg} x+\arcsin \frac{2 x}{x^{2}+1}, x \in \mathbb {R}$.

$y = (x-3)^{2} e^{\left|x\right|}, a = -1, b = 4$;

$y = (x-3)^{3} e^{\left|x+1\right|}, a = -2, b = 4$;

$y = e^{\sqrt{x^{2}\left|x+1\right|}}, a = -2, b = 1$;

$y = \ln \left(1+\sqrt{\left|x\right|(x+1)^{2}}\right), a = -2, b = 1$;

$y = \operatorname {arctg} \sqrt{\left|x\right|(x-1)^{2}}, a = -1, b = 2$;

$y = \operatorname {arcctg} \sqrt{x^{2}\left|1-x\right|}, a = -1, b = 2$.

$y(x) = \begin{cases} -x^{2}, & x \leqslant 0, \\ 2 e x \ln x, & x > 0,\end{cases} \quad a = -1, \quad b = 2$;

$y = \begin{cases} 1+3 x, & x \leqslant 0, \\ (x)^{x^{2}}, & x > 0,\end{cases} \quad a = -1, \quad b = 2$

$y = \left[x\right] / x+x / 3, a = 1, b = 3$.

$y = \cos (x+\pi / 3 \operatorname {sign} x)+\sin (x+\pi / 6), a = -\pi , b = \pi$;

$y = \sin (x-\pi / 4)-\cos (x-(3 \pi / 4) \operatorname {sign} x), a = -\pi , b = \pi$;

$y = \sin (x-\pi / 3)-\cos (x-(2 \pi / 3) \operatorname {sign} x), a = -\pi , b = \pi$.

$\left\{ 105 n+3 n^{2}-n^{3}\right\}$;

$\left\{ \frac{\sqrt{n}}{n+1985}\right\}$;

$\left\{ \frac{\sqrt[3]{n}}{n+19}\right\}$;

$\left\{ \frac{n^{2}}{n^{3}+200}\right\}$;

$\left\{ \frac{n^{12}}{e^{n}}\right\}$;

$\left\{ \frac{n^{10}}{2^{n}}\right\}$.

$f = 1 / x+x^{2}, x \in (0 ; 1]$;

$f = \ln x-x, x \in (0 ;+\infty )$;

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

$f = \operatorname {tg} x-3 x, x \in [-\pi / 4 ; \pi / 2)$.

$f = \left(x^{2}+4\right) e^{-x}, x \in (0 ;+\infty )$;

$f = e^{-x^{2}} \cos x^{2}, x \in \mathbb {R}$;

$f = (1 / x) e^{-1 / x}, x \in (0 ;+\infty )$;

$f = x^{x}, x \in (0 ; 1 / 2]$.

$f = x+\left(\frac{2}{x-2}\right)^{2}, x \in \left(\frac{3}{2} ; 5\right)$;

$f = (x+1)\left(\frac{x-1}{x-2}\right)^{2}, x \in \left(\frac{3}{2}; 6\right)$;

$f = \frac{\sqrt{1+\left|x-2\right|}}{1+\left|x\right|}, x \in \mathbb {R}$;

$f = \frac{x^{2}+1}{x^{2}+3}, x \in (a ;+\infty ), a \in \mathbb {R}$.

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

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

$f = e^{x}$

$f = \ln x$

$f = x \ln x$

$f = \operatorname {arctg} x$.

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

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

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

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

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