Question

Consider the following statements : 1. The function f(x) = |x| is not differentiable at x = 0. 2. The function $$f(x)=e^x$$ is differentiable at x = 0. Which of the above statements is/are correct ?

A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2

Solution

The correct option is C Both 1 and 2Statement 1$$f(x)=x \ for \ x \geq 0$$$$f(x)=-x \ for \ x < 0$$For a function to be differentiable at a point, it's derivative must be continuous at that point$$\Rightarrow$$ Left hand derivative(L.H.D)=Right hand derivative (R.H.D)$$L.H.D= f^{'}(x) \ for \ x<0$$$$\Rightarrow L.H.D=-1$$$$R.H.D=f^{'}(x) \ for \ x \geq 0$$$$\Rightarrow R.H.D =1$$$$\Rightarrow L..H.D \neq R.H.D$$ The function $$f(x)=\lvert x \rvert$$ is not differentiable at $$x=0$$Statement 2Derivative of function $$f(x)=e^x \ , f^{'}(x)=e^x$$$$f^{'}(x)\ is\ always\ continuous \ for \ all \ x \in R$$$$\Rightarrow$$ The function $$f(x)=e^x$$ is differentiable at $$x=1$$Both statement 1 and 2 are correct.Mathematics

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