Question

If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then
(a) f' (a⁺) = ϕ (a)
(b) f' (a⁻) = −ϕ (a)
(c) f' (a⁺) = f' (a⁻)
(d) none of these

Answer 1

(a) $f\text{'}\left(a^{+}\right)=\varphi \left(a\right)$
(b) $f\text{'}\left(a^{-}\right)=-\varphi \left(a\right)$

Here, $f\left(x\right)=\left|x-a\right|\varphi \left(x\right)$

$f\text{'}\left(a^{+}\right)=\underset{\mathrm{h}\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}=\underset{\mathrm{h}\to 0}{\lim }\frac{\left|h+a-a\right|\varphi \left(a+h\right)-\left|a-a\right|\varphi \left(a\right)}{h}=\underset{\mathrm{h}\to 0}{\lim }\frac{h\varphi \left(a+h\right)}{h}=\underset{\mathrm{h}\to 0}{\lim }\varphi \left(a+h\right)=\varphi \left(a\right)$

Also,

$f\text{'}\left(a^{-}\right)=\underset{\mathrm{h}\to 0}{\lim }\frac{f\left(a-h\right)-f\left(a\right)}{h}=\underset{\mathrm{h}\to 0}{\lim }\frac{\left|a-h-a\right|\varphi \left(a-h\right)-\left|a-a\right|\varphi \left(a\right)}{h}=\underset{\mathrm{h}\to 0}{\lim }\frac{\left|-h\right|\varphi \left(a-h\right)}{h}=\underset{\mathrm{h}\to 0}{\lim }-\varphi \left(a-h\right)=-\varphi \left(a\right)$