Finding the Greatest and Least Values of Functions of One Variable

Find the maximum of the function $y = 6x-x^{2}$.

Find the minimum of the function $y = x^{2}-8x$.

Find the extreme values of the function $y = x^{3}-12x$.

Show that the function $y = x^{3}(8-x)$ has a maximum at $x=6$ and has no extremum at $x=0$.

Show that the function $y = x^{2}+\dfrac {1}{2}x^{2}\sin \dfrac {1}{x}$ has a minimum at $x=0$, although the first derivative, as $x$ passes through zero, changes sign neither from $-$ to $+$ nor from $+$ to $-$.

Find the extreme values of the function $y = x^{3}-3x^{2}+6x+7$.

Find the extreme values of the function $y = x^{3}-9x^{2}+15x-3$.

Find the extreme values of the function $y = a+(x-b)^{4}$.

Find the extreme values of the function $y = a+(x-b)^{3}$.

Find the extreme values of the function $y = x^{4}-8x^{3}+22x^{2}-24x+12$.

Find the extreme values of the function $y = x^{5}-5x^{4}+5x^{3}-1$.

Find the extreme values of the function $y = (x-4)^{4}(x+3)^{3}$.

Find the extreme values of the function $y = \dfrac {x}{1+x^{2}}$.

Find the extreme values of the function $y = x+\dfrac {1}{x}$.

Find the extreme values of the function $y = \dfrac {a^{2}}{x}+\dfrac {b^{2}}{a-x}$.

Find the extreme values of the function $y = \dfrac {x^{2}-7x+6}{x-10}$.

Find the extreme values of the function $y = \dfrac {x^{4}+1}{x^{2}}$.

Find the extreme values of the function $y = \dfrac {x^{2}}{x^{4}+4}$.

Find the extreme values of the function $y = x\ln x$.

Find the extreme values of the function $y = x^{2}\ln x$.

Find the extreme values of the function $y = x^{x}$.

Find the extreme values of the function $y = x\ln^{2} x$.

Find the extreme values of the function $y = x^{n}e^{-x}$, $n$ positive.

Find the extreme values of the function $y = x^{2}e^{-x^{2}}$.

Find the extreme values of the function $y = e^{x}+e^{-x}$.

Find the extreme values of the function $y = e^{-x}-e^{-2x}$.

Find the extreme values of the function $y = x^{3}\sqrt[3]{(x-1)^{2}}$ for $-2 \leq x \leq 2$.

Find the extreme values of the function $y = \ln x-\arctg x$.

Find the extreme values of the function $y = e^{x}\cos x$.

Find the extreme values of the function $y = \dfrac {\sin x}{1-a^{2}\cos^{2}x}$ for $a^{2} < 1$ (consider two cases: $2a^{2} > 1$ and $2a^{2} < 1$).

Find the extreme values of the function $y = \dfrac {\tg x}{\tg (x+a)}$ for $0 < a < \dfrac {\pi }{2}$.

Find the extreme values of the function $y = \dfrac {\tg^{3} x}{\tg 3x}$.

Find the extreme values of the function $y = \sin 3x-3\sin x$.

Find the extreme values of the function $y = \arcsin \sin x$.

Find the extreme values of the function $y$ given implicitly by the equation $y^{2}+2yx^{2}+4x-3=0$.

Find the extreme values of the function $y$ given implicitly by the equation $y^{2}+2yx^{2}-4x-3=0$.

Find the extreme values of the function $y$ given implicitly by the equation $xy^{2}-x^{2}y=2a^{3}$.

Find the extreme values of the function $y$ given implicitly by the equation $x^{2}+2xy+y^{2}-4x+2y-2=0$.

Find the extreme values of the function $y$ given implicitly by the equation $x^{2}-2xy+5y^{2}-2x+4y+1=0$.

Find the extreme values of the function $y$ given implicitly by the equation $x^{2}+4xy+4y^{2}+x+2y-1=0$.

Find the extreme values of the function $y$ given implicitly by the equation $x^{3}+y^{3}-3axy=0$ for $a > 0$.

Find the extreme values of the function $y$ given implicitly by the equation $x^{4}+y^{4}-4xy=0$.

Find the extreme values of the function $y$ given implicitly by the equation $x^{4}+y^{4}=x^{2}+y^{2}$.