Question

Find the values of at which the function() • has a possible relative maximum or none of these minimum * There are no relative maximum maximum at

Answer 1

If f is a differentiable function, and x_0 is a critical point (local maximum or minimum), then f ' (x_0)=0.

To find the values of x_0 such that f ' (x_0)=0, differentiate the function:

$f^{\prime}(x)=\frac{d}{dx}f(x)=\frac{d}{dx}(e^{-2x}+2x)=-2e^{-2x}+2$

Assume that f ' (x_0)=0:

$f^{\prime}(x_0)=-2e^{-2x_0}+2=0_{}$

Isolate x_0:

$x_0=-\frac{1}{2}\ln (1)$

Since the natural logarithm of 1 is 0, then:

$x_0=0$

To check whether x_0 is a local maximum or a minimum, find the second derivative of f:

$f^{\prime\prime}(x)=\frac{d}{dx}f^{\prime}(x)=\frac{d}{dx}(-2e^{-2x}+2)=4e^{-2x}$

Evaluate f '' (x) at x=0:

$f^{\prime}^{\prime}(0)=4e^{-2(0)}=4e^0=4\cdot1=4$

Since f ' (0)=0 and f '' (0)=4>0, then 0 is a local minimum value for the function f.