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

The correct option is C Both 1 and 2
Statement 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 2

Derivative 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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