find point of inflection and extrema of y = x^5/3 -5x^2/3

Answer 1

Step 1:

The function is .

Apply derivative on each side with respect to .

.

Step 2:

Find the relative extrema by equating first derivative to zero.

Apply zero product property.

and

and

and

and .

Substitute in .

.

The point is

Substitute in .

.

The point is .

Comments

Contd...

Step 3:

Find the nature of relative extrema, using second derivative test.

Apply derivative on each side with respect to .

.

Point
sign of
Conclusion	Relative maximum	Relative minimum

The relative maximum at .

The relative minimum at .

Step 4:

Find the inflection points by equating the second derivative to zero.

.

Substitute   in .

.

The inflection point is .

Solution:

The relative maximum at .

The relative minimum at .

The inflection point is .