Question

Find the relative maxima and relative minima of f(x) = x³/3 - 25x⁵.

a) Relative maxima at x = -√3, relative minima at x = √3
b) Relative maxima at x = -√5, relative minima at x = √5
c) Relative maxima at x = -5, relative minima at x = 5
d) Relative maxima at x = -3, relative minima at x = 3

Answer 1

The relative maxima and minima of the function f(x) = x³/3 - 25x⁵ are x = -√3 and x = √3 respectively.

Step-by-step explanation:

To find the relative maxima and relative minima of the function f(x) = x³/3 - 25x⁵, we need to find the critical points by finding where the derivative equals zero. Taking the derivative of f(x), we get f'(x) = x² - 125x⁴. Setting this equal to zero and solving for x, we find the critical points at x = -√3 and x = √3. To determine whether these points are relative maxima or minima, we can use the second derivative test.

The second derivative of f(x) is f''(x) = 2x - 500x³. Plugging in the critical points, we find that f''(-√3) = -4√3 and f''(√3) = 4√3. Since the second derivative is negative at x = -√3, this means there is a relative maximum at x = -√3. And since the second derivative is positive at x = √3, this means there is a relative minimum at x = √3.