Question

List the functions which have relative extrema on the interval [-1,1]. Specify whether it is a local minimum or local maximum, and the value of x at which it occurs

Answer 1

Relative extrema are the peaks and valleys of a function's graph, with peaks representing relative maxima and valleys representing relative minima. The relative extrema are the combination of relative maxima and minima.

How do we find the relative extrema?

Assuming the function given is:

g(x) = x³ + 4x²

The x-coordinate of the relative maximum on the interval (-1,1) is


What is the explanation for the above?

To derive the relative maximums, we need to find where our first derivative changes sign. To do this, find your first derivative and then find where it is equal to zero.

Hence,

g(x) = x³ + 4x²

g'(x) = 3x² + 8x

g'(x) = x(3x + 8)

g'(x) = 0

Thus, at x = 0 = -(8/3)

This means that we have extrema at x = 0 and -8/3

Since we only need about the interval from -1 to 1, we only need to test points on that interval. Test points between our two extrema, as well as one between -8/3 and -5.

Hence,

g′(−1)=3∗(−12)+8∗−1=3+−8=−5

g' (-4) = 3 * (-4)² + (8* -4) = 48- 32 = 16

Hence, at -8/3, our first derivative transitions from negative to positive, and this is the x coordinate of our relative maximum on this interval.

Learn more abut relative extrema:
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