Question

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

• A. has no local minima
• B. has no local maxima
• C. is strictly increasing on R
• D. is strictly decreasing on R

Answer 1

Answer: (A), (B), (C)

The correct options are
A has no local minima
B is strictly increasing on R
C has no local maxima

$f\left(x\right)=x^3+b{x}^2+cx+d$

Differentiating w.r.t. $x$,

$f^{\prime }\left(x\right)=3x^2+2bx+c$ ..................... (1)

Discriminant for the above quadratic equation, $D=(2b{)}^2-4(3\left)\right(c)$

$=4b^2-12c$

$0<b^2<c$ (given)

So,

$b^2-c<0$

Now,

$D=4b^2-4c-8c$

$=4(b^2-c)-8c$

$\Rightarrow D<0$

Coefficient of $x^2$ in equation (1) is positive and $D<0$

So, $f^{\prime }\left(x\right)>0$ for any real value of $x$

Therefore, $f\left(x\right)$ is strictly increasing on $R$

and it has no local maxima or minima, because $f^{\prime }\left(x\right)\ne 0$ at any point on $R$