Question

Find the value of a for which the function f (x) = ax³-3(a+2)x²+9(a+2)x-1 is decreasing for all x ∈ R.

Answer 1

f(x) = ax³ - 3(a + 2)x² + 9(a + 2)x - 1 is decreasing for all x ∈ R.

so, f'(x) < 0 for all x ∈ R

so, first of all differentiate f(x) with respect to x,

i.e., f'(x) = 3ax² - 6(a + 2)x + 9(a + 2) < 0

concept : px² + qx + r < 0 is possible only if p < 0 or, D = q² - 4pr < 0

similarly, 3ax² - 6(a + 2)x + 9(a + 2) < 0

only if , 3a < 0 or, D = {-6(a + 2)}² - 4{9(a + 2)}3a < 0

a < 0 .....(1)

or, 36(a + 2)² - 108(a + 2)a < 0

⇒36(a + 2){(a + 2) - 3a } < 0

⇒(a + 2)(2 - 2a) < 0

⇒(a + 2)(1 - a) < 0

⇒ a > 1 or, a < -2 .....(2)

from inequalities (1) and (2),

a < -2

hence, for a ∈ (-∞,-2) , the function f(x) is decreasing for all x ∈ R.