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Let $f : R \to R$ be defined as $f (x) = {\begin{matrix} \frac{a - b \cos 2 x}{x^{2}} & ; & x < 0 \\ x^{2} + c x + 2 & ; & 0 \leq x \leq 1 \\ 2 x + 1 & ; & x > 1 \end{matrix}$
If f is continuous everywhere in R and m is the number of points where f is NOT differential then $2 m + a + b + c$ equals

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Detailed Solution

$f (x) = {\begin{matrix} \frac{a - b \cos 2 x}{x^{2}} & x < 0 \\ x^{2} + c x + 2 & 0 \leq x \leq 1 \\ 2 x + 1 & x > 1 \end{matrix}$

Continuous at $x = 0 \Rightarrow a = b & b = 1 ∴ a = b = 1$
Continuous at $x = 1 \Rightarrow c + 3 = 3 \Rightarrow c = 0$
Also $f (x)$ is differentiable for all pts $∴ m = 0, a = b = 1 c = 0$

$∴ R V = 2$

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