First slide

Monotonicity

Question

If $f (x) = {kx}^{3} - 9 x^{2} + 9 x + 3$ is monotonically increasing in R, then

NA

A

k<3
B

$k \leq 3$
C

$k \geq 2$
D

none of these

Solution

$f^{'} (x) = 3 {kx}^{2} - 18 x + 9 = 3 [{kx}^{2} - 6 x + 3] \geq 0 \forall x \in R or D = b^{2} - 4 ac \leq 0, k > 0, i.e., 36 - 12 k \leq 0 or k \geq 3$

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