Question

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function f(x)=x5−5x

Answer 1

The First Derivative Test can be used to determine the relative extreme values of the function f(x)=x^5−5x.

Taking the derivative of f(x) gives f'(x)=5x^4-5. Setting f'(x)=0 and solving for x gives x=1 and x=-1. These values divide the number line into three intervals: (-∞,-1), (-1,1), and (1,∞).

Testing the sign of f'(x) in each interval shows that f'(x) is negative on (-∞,-1), positive on (-1,1), and negative on (1,∞). This indicates that f(x) has a local maximum at x=-1 and a local minimum at x=1.