Question

Find the intervals in which the following functions are strictly increasing or decreasing: (a) x^2 + 2x − 5 (b) 10 − 6x − 2x^2 (c) −2x^3 − 9x^2 − 12x + 1 (d) 6 − 9x − x^2 (e) (x + 1)^3 (x − 3)^3

Answer 1

(a)Let f(x) = x² + 2x - 5
differentiate with respect to x,
f'(x) = 2x + 2
for increasing ,f '(x) > 0
=> 2x + 2 > 0 => x > -1 e.g., x∈(-1,∞)
for decreasing , f'(x) < 0
=> 2x + 2 < 0 => x < -1 e.g., x∈ (-∞, -1)


(b) Let f(x) = 10 - 6x - 2x²
differentiate with respect to x,
f'(x) = -6 - 4x
for increasing , f'(x) > 0
=> -6 - 4x > 0 => x > -3/2 e.g., x∈ (-3/2, ∞)
for decreasing , f'(x) < 0
=> -6 - 4x < 0 => x < -3/2 e.g., x∈(-∞, -3/2)


(c) Let f(x) = -2x³ -9x² -12x +1
differentiate with respect to x,
f'(x) = -6x² - 18x - 12 = -6(x² +3x + 2)
f'(x) = -6(x + 2)(x + 1)
for increasing, f'(x) > 0
=> -6(x + 2)(x + 1) > 0
=> x > -1 or, x < -2 e.g., x∈(-1,∞) U(-∞, -2)
for decreasing , f'(x) < 0
=> -6(x + 1)(x + 2) < 0
=> -2 < x < -1 e.g., x∈ (-2,-1)

(d) Let f(x) = 9 - 6x - x²
differentiate with respect to x,
f'(x) = -6 - 2x
for increasing , f'(x) > 0
=> -6 - 2x > 0 => x > -3 e.g., x∈(-3,∞)
for decreasing , f'(x) < 0
=> -6 - 2x < 0 => x < -3 e.g., x∈ (-∞, -3)

(e) Let f(x) = (x +1)³(x -3)³
differentiate with respect to x,
f'(x) = (x+1)³.3(x -3)² + (x -3)³ 3(x +1)²
= 3(x+1)²(x-3)² [ (x + 1) + (x - 3)]
= 6(x +1)²(x -3)² (x -1)
for increasing, f'(x) > 0
=> 6(x +1)² (x - 3)² (x -1) > 0
use number line concept,
then, f'(x) > 0 only when x∈ (1,3) U(3, ∞)
for decreasing , f'(x) < 0
3(x+1)²(x-3)²(x-2) < 0
use number line concepts ,
then, f'(x) < 0 only when x∈ (-∞ , -1) U (-1,1)