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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
Assertion (A): Let $f(x)=x^3+a x^2+b x+5 \sin ^2 x$, then the condition that $f(x)$ is always one-one function is $a^2-3 b+15<0$.
Reason (R) : $f(x)$ to be one one either $f$ is strictly increasing or strictly decreasing.
  • A
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • B
    Both (A) and (R) are true but (R) is not the correct explanation of (A).
  • C
    (A) is true but (R) is false.
  • D
    (A) is false but (R) is true.
✓

Answer

$
\begin{array}{l}
\text { (a): } f(x)=x^3+a x^2+b x+5 \sin ^2 x \\
\therefore \quad f^{\prime}(x)=3 x^2+2 a x+b+5 \sin 2 x
\end{array}
$
For one-one function, $f^{\prime}(x)>0$ for $x \in R$
$
\begin{array}{l}
\Rightarrow \quad 3 x^2+2 a x+b+5 \sin 2 x>0 \\
\Rightarrow \quad 3 x^2+2 a x+(b-5)>0 \Rightarrow(2 a)^2-4 \cdot 3(b-5)<0 \\
\Rightarrow \quad a^2-3 b+15<0
\end{array}
$

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