First Derivative Test for Local Maxima and Minima

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

$\mathrm{f}\left(\mathrm{x}\right)=\sin 2\mathrm{x}-\mathrm{x},-\frac{\mathrm{\pi }}{2}\le \mathrm{x}\le \frac{\mathrm{\pi }}{2}$

Find the points of local maxima or local minima of $\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{2}+\frac{2}{\mathrm{x}},\mathrm{x}>0$ using the first derivative test. Also, find the local maximum or minimum values.

Find the points of local maxima or minima of $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3{\left(2\mathrm{x}-1\right)}^3$ using the first derivative test. Also, find the local maximum or local minimum values.

Find the points of local maxima or minima of $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\sqrt{1-\mathrm{x}},\mathrm{x}>0$ using the first derivation test. Also, find the local maximum or local minimum values.

Find the points of local maxima or minima of $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2+2}$ using the first derivative test. Also, find the maximum and minimum values.

Find the point of local maxima or minima for $\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{x}-1\right){\left(\mathrm{x}+2\right)}^2$ using the first derivative test. Also, find the maximum and minimum value.

Find the points of local maxima or minima of $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3{\left(\mathrm{x}-1\right)}^2$ using the first derivative test. Also, find the local maximum or local minimum values.

Find the points of local maxima or minima for $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-3\mathrm{x}$ using the first derivative test. Also, find the local maximum or minimum value.

Find the points of local maxima or minima for $\mathrm{f}\left(\mathrm{x}\right)={\left(\mathrm{x}-5\right)}^4$ using the first derivative test. Also find the local maximum or minimum values.

Find the points of local maxima or local minima, if any, of the following functions, using the first derioative test. Also, find the local maximum or local minimum values, as the case may be:

Find the points of local maxima or local minima, if any, of the following functions, using the first derioative test. Also, find the local maximum or local minimum values, as the case may be:

$\mathrm{f}\left(\mathrm{x}\right)=2\mathrm{sinx}-\mathrm{x},-\frac{\mathrm{\pi }}{2}\le \mathrm{x}\le \frac{\mathrm{\pi }}{2}$

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection, if any:

$f\left(x\right)=x^3-6x^2+9x+15$

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

$f\left(x\right)=\sin x-\cos x,0<x<2\mathrm{\pi }$

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection, if any:

$f\left(x\right)=\sin 2x, 0<x<\mathrm{\pi }$