Increasing & Decreasing functions

The derivative of a function is positive over the range where it is increasing and negative where it is decreasing.

If y is increasing then, \( \frac{dy}{dx} \scriptsize > 0 \) 

If y is decreasing then, \( \frac{dy}{dx} \scriptsize < 0 \)

Example:

Solution:

Given y = 2x³ + 3x² -12x + 5

then \( \frac{dy}{dx} = \scriptsize 6x^2 + 6x\; – 12 \)

= 6(x² + x – 2)

= 6(x + 2)(x – 1)

The function is increasing when, \( \frac{dy}{dx} \scriptsize > 0 \), 

(x + 2)(x – 1) > 0

x > 1 or x < – 2

so the function is increasing when x > 1 or x < -2.