If f(x) = e^{|x|}, then which one of the following is correct?

Correct Answer is : f'(0) does not exist.

Correct Answer is : f'(0) does not exist.

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UPSC NDA I 2021 Mathematics Solved Paper

Question : 99 of 120

Marks: +1, -0

f(x) = e^{|x|}

, then which one of the following is correct?

Solution:

Concept:

Differentiability of

f(x) = e^{|x|}

x = 0

depends on equality of left and right derivatives.

Explanation:

|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}

f(x) = e^{x}

for

x \ge 0

and

f(x) = e^{-x}

for

x < 0

Right-hand derivative at

0

\lim_{h \to 0^+} \frac{e^{h} - e^{0}}{h} = \lim_{h \to 0^+} \frac{e^{h} - 1}{h} = 1

Left-hand derivative at

0

\lim_{h \to 0^-} \frac{e^{-h} - e^{0}}{h} = \lim_{h \to 0^-} \frac{e^{-h} - 1}{h} = -1

(since

e^{-h} \approx 1 - h

for small

h

Since the left-hand derivative (

-1

) and right-hand derivative (

1

) are not equal, the derivative at

x=0

does not exist.

Answer:

f'(0)

does not exist.

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