How do you use the second derivative test how do you find the local maxima and minima of #f(x) = 12 + 2x^2 - 4x^4#?

Question

Isaiah Allen · Accepted Answer

The function #f(x)=12+2x^2-4x^4# has derivative #f'(x)=4x-16x^3# and second derivative #f''(x)=4-48x^2#. The critical points occur where #f'(x)=4x(1-4x^2)=0#, which are #x=0# and #x=\pm 1/2#. Since #f''(\pm 1/2)=4-48\cdot 1/4=4-12=-8<0#, the second derivative test says the critical points at #x=\pm 1/2# are local maxima (the graph of #f# is concave down near #x=\pm 1/2#). Since #f''(0)=4>0#, the second derivative test says the critical point at #x=0# is a local minimum (the graph of #f# is concave up near #x=0#).

Cameron Alexander · Answer

undefined To use the second derivative test to find the local maxima and minima of $ f(x) = 12 + 2x^2 - 4x^4 $:

1. Find the first derivative of $ f(x) $ and set it equal to zero to find critical points.
2. Find the second derivative of $ f(x) $.
3. Evaluate the second derivative at each critical point.
4. If the second derivative is positive at a critical point, the function has a local minimum at that point. If the second derivative is negative, the function has a local maximum at that point. If the second derivative is zero or undefined, the test is inconclusive.