Question

If $f\left(x\right)=k{x}^3-9x^2+9x+3$ is monotonically increasing in each interval, then

• A. $k<3$
• B. $k\le 2$
• C. $k\ge 3$
• D. None of the above.

Answer 1

The correct option is C $k\ge 3$

The given function is:

$f\left(x\right)=k{x}^3-9x^2+9x+3$

For the function to be monotonically increasing we should have $f^{\prime }\left(x\right)>0$ over the domain of the function.

So differentiating once w.r.t to x we get,

$\Rightarrow f^{\prime }\left(x\right)=3k{x}^2-18x+9$

$\Rightarrow f^{\prime }\left(x\right)>0$

$\Rightarrow 3k{x}^2-18x+9>0$

For the quadratic equation to be always greater than zero the roots of the equation must be complex i.e. $b^2-4ac⩽0$

$\Rightarrow {18}^2-4\times 3k\times 9⩽0$

$\Rightarrow 324-108k⩽0$

$\Rightarrow 3-k⩽0$

$\Rightarrow k\ge 3$ .......Answer