Question

Let $f\left(x\right)=x^3+b{x}^2+cx+d,0<b^2<c$. then $f$:

• A. Is bounded
• B. Has a local maxima
• C. Has a local minima
• D. Is strictly increasing

Answer 1

The correct option is D

Is strictly increasing

Explanation for the correct answer:

Given that: $f\left(x\right)=x^3+b{x}^2+cx+d,0<b^2<c$.

Differentiating the given function with respect to $x$, we get: $f\text{'}\left(x\right)=3x^2+2bx+c$.

Thus, the discriminant of $f\text{'}\left(x\right)=3x^2+2bx+c$ is $D=4\left(b^2-3c\right)$.

Given that: $0<b^2<c$. Then $b^2-3c<0$.

Therefore, the discriminant $D=4\left(b^2-3c\right)$ will be negative.

Hence, $f\text{'}\left(x\right)$ does not have real roots.

So, $f\text{'}\left(0\right)>0$ for all $x\in R$.

Therefore, $f$ is strictly increasing.

Hence, option (D) is the correct answer.