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

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

Concept:

Differentiability of a function:

We will define the differentiability of the function with the help of its graph. If a graph of a function is smooth everywhere then the function is said to be differentiable everywhere in the xy - plane.

If the function has a sharp edge or a vertical asymptote (tangent) to the curve then the function is not differentiable at that point.

 

Calculation:

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Observe the graph of the function f(x) = |x|.

At x = 1 the curve is smooth and thus the function is differentiable at x = 1.

On the other hand the function is not differentiable at x = 0 as it has a corner at x = 0.

 

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Observe the graph of the function f(x) = ex.

At x = 0 the curve is smooth so it is differentiable at x = 0.

Thus only the second statement is true.

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