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  • Let <img alt="f(x)= \begin{cases}x^p \sin \left(\frac{1}{x}\right)+x\left|x^3\right|, & x \neq 0 \\ 0, & x=0\end{cases}" src="https://entrancecorner.oncodecogs.com/gif.latex?f%28x%29%3

Let f(x)= \begin{cases}x^p \sin \left(\frac{1}{x}\right)+x\left|x^3\right|, & x \neq 0 \\ 0, & x=0\end{cases} 

the set of values, of p for which f ′′(x) is continuous at x = 0 is

Option: 1

[2, ∞)


Option: 2

 [–3, ∞)


Option: 3

[5, ∞)


Option: 4

none of these

 


Answers (1)

best_answer

f(x)= \begin{cases}x^p \sin \left(\frac{1}{x}\right)+x^4, & x>0 \\ x^p \sin \left(\frac{1}{x}\right)-x^4, & x<0 \\ 0, \quad x=0 & \end{cases}

\Rightarrow f^{\prime}(x)= \begin{cases}-x^{p-2} \cos \left(\frac{1}{x}\right)+p x^{p-1} \sin \left(\frac{1}{x}\right)+4 x^3, & x>0 \\ x^{p-2} \cos \left(\frac{1}{x}\right)+p x^{p-1} \sin \left(\frac{1}{x}\right)-4 x^3, & x<0 \\ 0, & x=0\end{cases}\therefore f^{\prime \prime}(x)= \begin{cases}-x^{p-4} \sin \left(\frac{1}{x}\right)-(p-2) x^{p-3} \cos \left(\frac{1}{x}\right)-p x^{p-3} \cos \left(\frac{1}{x}\right)+p(p-1) x^{p-2} \sin \left(\frac{1}{x}\right)+12 x^2, & x>0 \\ -x^{p-4} \sin \left(\frac{1}{x}\right)-(p-2) x^{p-3} \cos \left(\frac{1}{x}\right)-p x^{p-3} \cos \left(\frac{1}{x}\right)+p(p-1) x^{p-2} \sin \left(\frac{1}{x}\right)-12 x^2, & x<0 \\ 0, & x=0\end{cases}\thereforesin\infty and cos\infty lies between –1 to 1.

for \mathrm{p} \geq 5 RHL = 0

LHL = 0

V.F.  = 0

\text { for } p \in[5, \infty), f^{\prime \prime}(x) \text { is continuous . }

 

 

Posted by

jitender.kumar

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