If the function f(x)=x^3-9 k x^2+27 x+30 is increasing on R, then…

MCQ

If the function $f(x)=x^3-9 k x^2+27 x+30$ is increasing on $R$, then :

✓
$-1\leq\text{k}\leq1$
B
$k < -1$ or $k > 1$
C
$0 < k < 1$
D
$-1 < k < 0$

✓

Answer

Correct option: A.

$-1\leq\text{k}\leq1$

$f(x)=x^3-9 k x^2+27 x+30$
$\Rightarrow f^{\prime}(x)=3 x^3-18 k x+27$
$\Rightarrow 3\left(x^2-6 k x+9\right)$
Function is always increasing on $R$ .
$3\left(x^2-6 k x+9\right)>0$
$x^2-6 k x+9 > 0$
In $a x^2+b x+c=0 \text { if } a > 0$
$ \Rightarrow b^2-4 a c<0$
$36 k^2-36<0$
$k^2-1 < 0$
$(k+1)(k-1) < 0$
$\Rightarrow-1$

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