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If $f(x)=k x^{3}-9 x^{2}+9 x+3$ is monotonically increasing in every interval, then which one of the following is correct?
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Verified Answer
The correct answer is:
$k>3$
Given $f(x)=k x^{3}-9 x^{2}+9 x+3$ On differentiating w.r.t. $x$, we get $f^{\prime}(x)=3 k x^{2}-18 x+9$
For a function to be monotonically increasing. $b^{2}-4 a c < 0$
Here, $a=3 k, b=-18, c=9$
$\therefore^{2}-4 a c=(-18)^{2}-4(3 k)(9)=(-18)(-18)-(3 k) 18 \times 2$
$\Rightarrow 36-12 k < 0$
$\Rightarrow k>3$
For a function to be monotonically increasing. $b^{2}-4 a c < 0$
Here, $a=3 k, b=-18, c=9$
$\therefore^{2}-4 a c=(-18)^{2}-4(3 k)(9)=(-18)(-18)-(3 k) 18 \times 2$
$\Rightarrow 36-12 k < 0$
$\Rightarrow k>3$
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