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Definition of Functions

Let f: ℝ→ℝ be...

Question

Let $f : R \to R$ be defined as $f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x^{5} sin (\frac{1}{x}) & + 5 x^{2}, & x < 0 0 & x = 0 x^{5} cos (\frac{1}{x}) & + λ x^{2}, & x > 0 \end{matrix}$ The value of $λ$ for which $f^{''} (0)$ exists, is

A

5

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B

5.00

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C

5.0

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Solution

$f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x^{5} sin (\frac{1}{x}) + 5 x^{2}, & x < 0 0 & x = 0 x^{5} cos (\frac{1}{x}) + λ x^{2}, & x > 0 \end{matrix}$

$f^{'} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 5 x^{4} sin (\frac{1}{x}) - x^{3} cos (\frac{1}{x}) + 10 x, & x < 0 0 & x = 0 5 x^{4} cos (\frac{1}{x}) + x^{3} sin (\frac{1}{x}) + 2 λ x, & x > 0 \end{matrix}$

$f^{''} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 20 x^{3} sin (\frac{1}{x}) - 5 x^{2} cos (\frac{1}{x}) - 3 x^{2} cos (\frac{1}{x}) - x sin (\frac{1}{x}) + 10, & x < 0 0 & x = 0 20 x^{3} cos (\frac{1}{x}) + 5 x^{2} sin (\frac{1}{x}) + 3 x^{2} sin (\frac{1}{x}) - x cos (\frac{1}{x}) + 2 λ, & x > 0 \end{matrix}$
if $f^{''} (0)$ exists then
$f^{''} (0^{+}) = f^{''} (0^{-})$
$2 λ = 10 \Rightarrow λ = 5$

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