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)=(d)/(dx)f(x)=(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=-(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)=(d)/(dx)f^(\prime)(x)=(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.