AP EAMCET202419 May 2024Evening ShiftMathematicsContinuity and DifferentiabilityActual

If f(x)= array cc x^ ( 1 x ), & x 0 0, & x=0 array . ; Which of the following is true?

Options

Af(x) is continuous and differentiable if 0 1
Bf(x) is discontinuous and not differentiable if 0 1
Cf(x) is continuous and differentiable for 1
Df(x) is discontinuous and differentiable for 1

Correct answer

C. f(x) is continuous and differentiable for 1

Step-by-step solution

Given, f(x)= array cc x^ ( 1 x ) ; & x 0 0 ; & x=0 array . _ x 0 f(x)= _ x 0 x^ ( 1 x )=0 if 0 So, f(x) is continuous for 0 and f^ (0)= _ h 0 f(0+h)-f(0) h = _ h 0 h^ ( 1 h )-0 h = _ x 0 h^ -1 ( 1 h ) So f^ (0) is finite and exists if 1 f(x) is differentiable at x=0 if 1 .

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