Question

Find the intervals in which the function $f(x) = 5x^3 - 3x^2$ is (i) increasing (ii) decreasing.

Hint:

To find intervals of increase or decrease, analyze the sign of the first derivative and check where it is positive (increasing) or negative (decreasing).

Answer 1

To find the intervals of increase and decrease, we first compute the first derivative of \( f(x) \): \[ f'(x) = \frac{d}{dx}(5x^3 - 3x^2) = 15x^2 - 6x \] Now, find the critical points by setting the derivative equal to zero: \[ 15x^2 - 6x = 0 \implies x(15x - 6) = 0 \] So, \( x = 0 \) and \( x = \frac{2}{5} \) are the critical points.
To determine the intervals where the function is increasing or decreasing, we analyze the sign of \( f'(x) \):
- For \( x<0 \), \( f'(x)>0 \) (increasing).
- For \( 0<x<\frac{2}{5} \), \( f'(x)<0 \) (decreasing).
- For \( x>\frac{2}{5} \), \( f'(x)>0 \) (increasing).
Thus, the function is increasing in \( (-\infty, 0) \) and \( (0, \infty) \).