Applications Of Derivatives Question 45
Question: If $ f(x)=kx^{3}-9x^{2}+9x+3 $ is monotonically increasing in every interval, then which one of the following is correct-
Options:
A) $ k<3 $
B) $ k\le 3 $
C) $ k>3 $
D) $ k\ge 3 $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Given $ f(x)=kx^{3}-9x^{2}+9x+3 $ On differentiating w.r.t.x, we get $ f’(x)=3kx^{2}-18x+9 $ For a function to be monotonically increasing. $ b^{2}-4ac<0 $ Here, $ a=3k,b=-18,c=9 $
$ \therefore b^{2}-4ac={{(-18)}^{2}}-4(3k)(9) $
$ =(-18)(-18)-(3k)18\times 2 $
$ \Rightarrow 36-12k<0\Rightarrow k>3 $
BETA
