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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $f: R \rightarrow R$ be defined as

$f(x)=\left\{\begin{array}{ccc}x^{5} \sin \left(\frac{1}{x}\right)+5 x^{2}& , & x<0 \\ 0 & , & x=0 \\ x^{5} \cos \left(\frac{1}{x}\right)+\lambda x^{2} & , & x>0\end{array} .\right.$

The value of $\lambda$ for which $f^{\prime \prime}(0)$ exists, is

  • ✓
    $5$
  • B
    $10$
  • C
    $15$
  • D
    $20$
✓

Answer

Correct option: A.
$5$
a
$f(x)=x^{5} \cdot \sin \frac{1}{x}+5 x^{2} \quad$ if $x<0$

$f(x)=0$

if $x=0$

$f(x)=x^{5} \cdot \cos \frac{1}{x}+\lambda x^{2} \quad$ if $x>0$

LHD of $f^{\prime}(x)$ at $x=0$ is 10 $RHD$ of $f^{\prime}( x )$ at $x =0$ is $2 \lambda$

if $f^{\prime \prime}(0)$ exists then $2 \lambda=10$

$\Rightarrow \lambda=5$

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