Question

Let $λ^*$ be the largest value of $λ$ for which the function $f_λ(x) = 4λx^3 – 36λx^2 + 36x + 48$ is increasing for all $x ∈ ℝ$. Then $f_λ^* (1) + f_λ^* (– 1)$ is equal to :

• A. 36
• B. 48
• C. 64
• D. 72

Answer 1

Answer: (D)

72

Answer 2

As $f_λ(x) = 4λx^3 – 36λx^2 \+ 36x \+ 48$

Hence, $f_λ^{'}(x) = 12(λx^2-6λx+3)$

For $fλ(x)$increasing : $(6λ)^2 – 12λ ≤ 0$

$∴ λ ∈[0,\frac{1}{3}]$

$∴ λ ^∗=\frac{1}{3}$

Then,

$fλ^*(x) = \frac{4}{3}x^3−12x^2+36x+48$

$∴fλ^*+fλ^*(−1)=73\frac{1}{2}−1\frac{1}{2}$

= $72$.

Hence, the correct option is (D): $72$