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UPSC NDA I 2025 Mathematics Paper

Question Numbers: 79-80

Consider the following for the two (02) items that follow:

Letf(x)=\begin{cases} x^{3}, & x^{2}<1 \\ x^{2}, & x^{2}\ge 1 \end{cases}

Question : 79 of 120

Marks: +1, -0

What is

\underset{x→0}{lim}f^{′}(x)

equal to?

Solution:

Concept:

A piecewise function’s derivative near a point is determined by the rule valid in that neighbourhood. Here, close to

x=0

, the function behaves as

x^{3}

Explanation:

The function is

f(x)=\begin{cases} x^{3}, & |x|<1 \\ x^{2}, & |x|\ge 1 \end{cases}

. For

|x|<1

, the derivative is

f'(x)=3x^{2}

. Since the limit

x\to0

occurs entirely within this region, we evaluate

\displaystyle\lim_{x\to0}3x^{2}=0

. The limit exists and equals

0

Answer:

\displaystyle\lim_{x\to0}f'(x)=0

, i.e. option C.

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