Question

Find the intervals in which the following functions are strictly increasing or decreasing:

(a) x² + 2x − 5 

(b) 10 − 6x − 2x²
(c) −2x³ − 9x² − 12x + 1 

(d) 6 − 9x − x²
(e) (x + 1)³ (x − 3)³

Answer 1

(a) We have,

Point x = −1 divides the real line into two disjoint intervals i.e., (∞,-1) and (-1,∞)

Thus, f is strictly increasing for x > −1.

(b) We have,

f(x) = 10 − 6x − 2x²

The point x=3/2 divides the real line into two disjoint intervals

(c) We have,

f(x) = −2x³ − 9x² − 12x + 1

Points x = −1 and x = −2 divide the real line into three disjoint intervals

∴ f is strictly decreasing for x < −2 and x > −1.

The point x =9/2 divides the real line into two disjoint intervals i.e.,

The points x = −1, x = 1, and x = 3 divide the real line into four disjoint intervals