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,

Now,

x = −1

Point x = −1 divides the real line into two disjoint intervals i.e.,

In interval

∴f is strictly decreasing in interval

Thus, f is strictly decreasing for x < −1.

In interval

∴ f is strictly increasing in interval

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

(b) We have,

f(x) = 10 − 6x − 2x²

The pointdivides the real line into two disjoint intervals i.e.,

In interval i.e., when,

∴ f is strictly increasing for .

In interval i.e., when,

∴ f is strictly decreasing for .

(c) We have,

f(x) = −2x³ − 9x² − 12x + 1

Points x = −1 and x = −2 divide the real line into three disjoint intervals i.e.,

In intervals i.e., when x < −2 and x > −1,

.

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

Now, in interval (−2, −1) i.e., when −2 < x < −1, .

∴ f is strictly increasing for .

(d) We have,

The pointdivides the real line into two disjoint intervals i.e., .

In interval i.e., for, .

∴ f is strictly increasing for.

In interval i.e., for,

∴ f is strictly decreasing for.

(e) We have,

f(x) = (x + 1)³ (x − 3)³

The points x = −1, x = 1, and x = 3 divide the real line into four disjoint intervals i.e.,, (−1, 1), (1, 3), and.

In intervalsand (−1, 1), .

∴ f is strictly decreasing in intervalsand (−1, 1).

In intervals (1, 3) and, .

∴ f is strictly increasing in intervals (1, 3) and.