Question

Identifying ExtremaIn Exercises 19–40:a. Find the open intervals on which the function is increasing and those on which it is decreasing.b. Identify the function’s local extreme values, if any, saying where they occur.f(x) = x³ / (3x² + 1)

Accepted Answer

To determine where the function $$ f(x) = \frac{x^3}{3x^2 + 1}  is $$increasing or decreasing, first find the derivative $$ f'(x) . $$Use the quotient rule: $$ f'(x) = \frac{(3x^2)(3x^2 + 1) - (x^3)(6x)}{(3x^2 + 1)^2} . $$Simplify the expression to get the derivative. Set the derivative $$ f'(x) $$ equal to zero to find critical points. Solve $$ (3x^2)(3x^2 + 1) - (x^3)(6x) = 0  to $$find the values of $$ x $$ where the derivative is zero or undefined. Determine the sign of $$ f'(x)  on $$the intervals defined by the critical points. Choose test points in each interval and substitute them into $$ f'(x)  to $$see if the function is increasing ($$ f'(x) \u003e 0 ) or $$decreasing ($$ f'(x) \u003c 0 ). $$Identify the local extrema by analyzing the sign changes of $$ f'(x) . If$$ $$ f'(x) $$ changes from positive to negative at a critical point, there is a local maximum. If it changes from negative to positive, there is a local minimum. Finally, state the open intervals where the function is increasing or decreasing and identify any local extrema, specifying their locations based on the critical points and the behavior of $$ f'(x) .$$

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = x³ / (3x² + 1)

Verified video answer for a similar problem:

Key Concepts

Derivative and Critical Points

First Derivative Test

Interval Testing