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STD 12 - 6. Application of derivatives — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 6. Application of derivatives1 Mark
MCQ
Let $f(x)=\left\{\begin{array}{cc}x^{3}-x^{2}+10 x-7, & x \leq 1 \\ -2 x+\log _{2}\left(b^{2}-4\right), & x>1\end{array}\right.$ Then the set of all values of $b$, for which $f(x)$ has maximum value at $x=1$, is.
  • A
    $(-6,-2)$
  • B
    $(2,6)$
  • ✓
    $[-6,-2) \cup(2,6]$
  • D
    $[-\sqrt{6},-2) \cup(2, \sqrt{6}]$
✓

Answer

Correct option: C.
$[-6,-2) \cup(2,6]$
c
$f(1)=3$

For $x <1, f ^{\prime}( x )=3 x ^{2}-2 x +10>0$ $\Rightarrow f ( x )$ is increasing

For $x >1, f ^{\prime}( x )<0$

function is decreasing.

$\lim _{x \rightarrow 1^{+}} f(x)=-2+\log _{2}\left(b^{2}-4\right)$

For maximum value at $x=1$

$3 \geq-2+\log _{2}\left(b^{2}-4\right)$

$32 \geq b^{2}-4>0$

$b \in[-6,-2) \cup(2,6]$

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