# If f(x)= 12+2x^2-4x^4, use the second derivative test to identify the local maxima and minima.

## Question:

If {eq}f(x)= 12+2x^2-4x^4, {/eq} use the second derivative test to identify the local maxima and minima.

## Local Extrema:

We can find the extremes of a function in two ways, first using the first derivative test, where we study the intervals of increase and decrease and decide which of the critical points of the function is maximum and minimum. Another way is using the second derivative test. This consists of evaluating the critical point in the second derivative, if the result is positive (Minimum), if the result is negative (Maximum).

We have the function :

{eq}f(x)= 12+2x^2-4x^4 \\ {/eq}

Differentiating the function

{eq}f'(x)=-16\,{x}^{3}+4\,x \\ f''(x)=-48\,{x}^{2}+4 \\ {/eq}

{eq}f'(x)=0 {/eq} when {eq}x=0 \\ x=0.5 \\ x=-0.5 \\ {/eq}

Now,

{eq}f''(0)= 4 \\ f''(0.5)=-8 \\ f''(-0.5)=-8 \\ {/eq}

Therefore,

Local minimum at:

{eq}(0, 12 ) {/eq}

Local maximum at:

{eq}(0.5, 12.25) \\ (-0.5, 12.25) \\ {/eq} 