What are the local extrema, if any, of #f (x) = x^4-8x^2-48 #?
1 Answer
Local maximum at
Local minimum at
Explanation:
Local extrema occur when
Differentiating wrt
# f'(x)=4x^3-16x #
So if
We can now examine the nature of these turning points by looking at the second derivative:
Differentiating
# f''(x)=12x^2-16 #
so
And the values of the function at these extrema are:
#f(0) \ \ \ \ \ = 0 - 0 - 48 \ \ \ = -48#
#f(+-2) = 16 - 24 - 48 = -56#
Hence the extrema are:
Local maximum at
#(0,-48)#
Local minimum at#(-2,-56)# and#(2,-56)#
We can confirm this visually by plotting the graph:
graph{x^4-8x^2-48 [-10, 10, -74.2, 74]}