MCQ
If the function $f(x)=k x^3-9 x^2+9 x+3$ 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.
$\text{k}>3\text{k}>3$
$f(x)=k x^3-9 x^2+9 x+3$
$f^{\prime}(x)=k x^2-27$
$=3\left(x^2-9\right)$
For $f(x)$ to be increasing, we must have
$f^{\prime}(x)>0$
$\Rightarrow 3\left(x^2-9\right)>0$
$\Rightarrow\left(x^2-9\right)>0[$Since$, 3>0,3\left(x^2-9\right)>0 \Rightarrow\left(x^2-9\right)>0]$
$\Rightarrow(x+3)(x-3)>0$
$\Rightarrow x<-3$ or $x>3$
$\Rightarrow|x|>3$
$f^{\prime}(x)=k x^2-27$
$=3\left(x^2-9\right)$
For $f(x)$ to be increasing, we must have
$f^{\prime}(x)>0$
$\Rightarrow 3\left(x^2-9\right)>0$
$\Rightarrow\left(x^2-9\right)>0[$Since$, 3>0,3\left(x^2-9\right)>0 \Rightarrow\left(x^2-9\right)>0]$
$\Rightarrow(x+3)(x-3)>0$
$\Rightarrow x<-3$ or $x>3$
$\Rightarrow|x|>3$
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