Question

For what values of$a,f\left(x\right)=$ $-x^3+4a{x}^2+2x-5$is decreasing $\forall x$

• A. $(1,2)$
• B. $(3,4)$
• C. $R$
• D. No value of $a$

Answer 1

The correct option is D

No value of $a$

Explanation for the correct option:

Given that the function $f\left(x\right)=-x^3+4a{x}^2+2x-5$

Now, differentiate with respect to $x$,

$\Rightarrow f\text{'}\left(x\right)=–3x^2+8ax+2$

Since $–3x^2+8ax+2<0$

$\Rightarrow f\text{'}\left(x\right)<0$

Therefore $f\left(x\right)$is decreasing, all $x,$$f\text{'}\left(x\right)<0$

Thus, It is clear that no value of $a,f\left(x\right)$ is decreasing.

Hence, option(D) is correct.