If the function f(x) = kx^3 - 9x^2 + 9x + 3 is monotonically incr… - Vidyadip

MCQ

If the function $f(x) = kx^3 - 9x^2 + 9x + 3$ is monotonically increasing in every interval, then:

A
$\text{k}<3\text{k}<3$
B
$\text{k}\leq3\text{k}\leq3$
✓
$\text{k}>3\text{k}>3$
D
$\text{k}\geq3$

✓

Answer

Correct option: C.

$\text{k}>3\text{k}>3$

$f(x) = kx^3 - 9x^2 + 9x + 3$
$f'(x) = kx^2 - 27$
$= 3(x^2 - 9)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow 3(x^2 - 9) > 0$
$\Rightarrow (x^2 - 9) > 0$ $[$Since, $3 > 0, 3(x^2 - 9) > 0 \Rightarrow (x^2 - 9) > 0]$
$\Rightarrow (x + 3)(x - 3) > 0$
$\Rightarrow x < -3$ or $x > 3$
$\Rightarrow |x| > 3$

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