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Question: The function f defined by \(\frac{1}{2}\left\lbrack f\left( \frac{x}{y} \right) + f(xy) \right\rbrac...

The function f defined by 12[f(xy)+f(xy)]\frac{1}{2}\left\lbrack f\left( \frac{x}{y} \right) + f(xy) \right\rbrackis

A

Continuous and derivable at x = 0

B

Neither continuous nor derivable at x = 0

C

Continuous but not derivable at x = 0

D

None of these

Answer

Continuous and derivable at x = 0

Explanation

Solution

f ′ (0 + h) = lim⁡h→0\lim _ { h \rightarrow 0 }

= lim⁡h→0\lim _ { h \rightarrow 0 } = lim⁡h→0\lim _ { h \rightarrow 0 } = 1

f ′ (0 – h) = lim⁡h→0\lim _ { h \rightarrow 0 } f(0−h)−f(0)−h\frac { \mathrm { f } ( 0 - \mathrm { h } ) - \mathrm { f } ( 0 ) } { - \mathrm { h } }

= lim⁡h→0\lim _ { h \rightarrow 0 } sin⁡(−h)2−h−0−h\frac { \frac { \sin ( - h ) ^ { 2 } } { - h } - 0 } { - h } = lim⁡h→0\lim _ { h \rightarrow 0 } = 1