Let f ( x ) = x^3 + bx^2 + cx + d , 0 < b^2 < c , then f - Vidyadip

MCQ

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

A
is bounded
B
has a local maxima
C
has a local minima
✓
is strictly increasing

✓

Answer

Correct option: D.

is strictly increasing

d
$f(\mathrm{x})=\mathrm{x}^{3}+\mathrm{bx}^{2}+\mathrm{cx}+\mathrm{d}, 0<\mathrm{b}^{2}<\mathrm{c}$

$f^{\prime}(x)=3 x^{2}+2 b x+c.$

$D=4 b^{2}-12 c=4\left(b^{2}-3 c\right)=4\left(\left(b^{2}-c\right)-3 c\right)<0$

$\Rightarrow f^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in \mathrm{R}.$

Hence, $f(\mathrm{x})$ is strictly increasing.

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