Question

If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then

विकल्प

• f' (a⁺) = ϕ (a)

• f' (a⁻) = −ϕ (a)

• f' (a⁺) = f' (a⁻)

• none of these

Answer 1

\[f'\left( a^+ \right) = \phi\left( a \right)\]

\[f'\left( a^- \right) = - \phi\left( a \right)\]

Here, 

\[f\left( x \right) = \left| x - a \right| \phi\left( x \right)\]

\[f'\left( a^+ \right) = \lim_{h \to 0} \frac{f\left( a + h \right) - f\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| h + a - a \right|\phi\left( a + h \right) - \left| a - a \right|\phi\left( a \right)}{h} = \lim_{h \to 0} \frac{h \phi\left( a + h \right)}{h} = \lim_{h \to 0} \phi\left( a + h \right) = \phi\left( a \right)\]

Also,

\[f'\left( a^- \right) = \lim_{h \to 0} \frac{f\left( a - h \right) - f\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| a - h - a \right| \phi\left( a - h \right) - \left| a - a \right| \phi\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| - h \right| \phi\left( a - h \right)}{h} = \lim_{h \to 0} - \phi\left( a - h \right) = - \phi\left( a \right)\]