Question

If $f\left(x\right)=x^3+4x^2+\mathrm{kx}+1$ is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then .

• A. k = 4
• B. k = 2
• C. k = -1
• D. k has no real value

Answer 1

The correct option is A k = 4

$\mathrm{Here},f\text{'}\left(x\right)\le 0\mathrm{or},3x^2+8x+k\le 0,\forall x\in \left(-2,-\frac{2}{3}\right)$
Since, $3x^2+8x+k$ represents a upward parabola. Then , the situation for f '(x) are as follows:
Case1.

Case2.

Given that f(x) decreases in the largest possible interval $\left(-2,-\frac{2}{3}\right)$. Then,f '(x) must have roots -2 and -2/3.
$\mathrm{Thus},\mathrm{Product}\mathrm{of}\mathrm{roots}=\left(-2\right)\left(-\frac{2}{3}\right)=\frac{k}{3}\mathrm{or},k=4.$