Let f : R $\to$ R be defined as
$$f\left( x \right) = \left\{ {\matrix{
{{x^5}\sin \left( {{1 \over x}} \right) + 5{x^2},} & {x < 0} \cr
{0,} & {x = 0} \cr
{{x^5}\cos \left( {{1 \over x}} \right) + \lambda {x^2},} & {x > 0} \cr
} } \right.$$
The value of $\lambda$ for which f ''(0) exists, is _______.
Answer (integer)
5
Solution
If g(x) = x<sup>5</sup>sin$\left( {{1 \over x}} \right)$
<br><br>and h(x) = x<sup>5</sup>cos$\left( {{1 \over x}} \right)$
<br><br>then g''(0) = 0 and h''(0) = 0
<br><br>So, f''(0<sup>+</sup>
) = g''(0<sup>+</sup>
) + 10 = 10
<br><br>and f''(0<sup>–</sup>) = h''(0<sup>–</sup>) + 2$\lambda$ = f''(0<sup>+</sup>)
<br><br>$\Rightarrow$ 2$\lambda$ = 10
<br><br>$\Rightarrow$ $\lambda$ = 5
About this question
Subject: Mathematics · Chapter: Limits, Continuity and Differentiability · Topic: Limits and Standard Results
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