Find the intervals in which the function $f(x) = 5x^3 - 3x^2$ is (i) increasing (ii) decreasing.
Show 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).
The intervals of increase and decrease are found by first computing the first derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}(5x^3 - 3x^2) = 15x^2 - 6x
\]
Critical points are determined by setting the derivative to zero:
\[
15x^2 - 6x = 0 \implies x(15x - 6) = 0
\]
The critical points are \( x = 0 \) and \( x = \frac{2}{5} \).
The sign of \( f'(x) \) is analyzed to determine intervals of increase and decrease:
For \( x<0 \), \( f'(x)>0 \), indicating the function is increasing.
For \( 0<x<\frac{2}{5} \), \( f'(x)<0 \), indicating the function is decreasing.
For \( x>\frac{2}{5} \), \( f'(x)>0 \), indicating the function is increasing.
Therefore, the function increases in the intervals \( (-\infty, 0) \) and \( (\frac{2}{5}, \infty) \).