First slide

Monotonicity

Question

lf $f (x) = x^{3} + 4 x^{2} + λx + 1$ is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then

Moderate

A

$λ = 4$
B

$λ = 2$
C

$λ = - 1$
D

$λ$ has no real value

Solution

Here, $f^{'} (x) \leq 0$
or $3 x^{2} + 8 x + λ \leq 0 \forall x \in (- 2, - \frac{2}{3})$
Then situations form f '(x) are as follows:

Given that /(*) decreases in the largest possible interval $(- 2, - \frac{2}{3})$ .

Then $f' (x) = 0$ must have roots -2 and -2/3.
Thus, product roots $= (- 2) (- \frac{2}{3}) = \frac{λ}{3} or λ = 4$

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