If p(x)=f(x) g(x), then which of the following statements is/are correct? I. p(x) is continuous at x=0. II. p(x) is differentiable at x=0. Select the answer using the code given below :

Correct Answer is : Both I and II

Correct Answer is : Both I and II

Test Index

UPSC NDA I 2026 Mathematics Paper

Question Numbers: 99-100

For the next two (02) items that follow:

Let

f(x)=\tan \left(x^{2}\right)

and

g(x)=x|x| \text { for }|x| < \sqrt{\frac{\pi}{2}}

Question : 99 of 120

Marks: +1, -0

p(x)=f(x) g(x)

p(x)

x=0

p(x)

x=0

Select the answer using the code given below :

Solution:

Concept:

Check continuity and differentiability of

p(x)=f(x)g(x)

x=0

using limits.

Explanation:

Given:

f(x)=\tan(x^2)

and

g(x)=x|x|

So,

p(x)=x|x|\tan(x^2)

x=0

p(0)=0 \cdot |0| \cdot \tan(0)=0

For continuity at

x=0

\lim_{x \to 0} p(x)=\lim_{x \to 0} x|x|\tan(x^2)

Since

\tan(x^2) \to 0

x \to 0

, and

x|x| \to 0

the product tends to

0

So,

p(x)

is continuous at

x=0

For differentiability at

x=0

\lim_{h \to 0} \frac{p(h)-p(0)}{h} < br> < br>= \lim_{h \to 0} \frac{h|h|\tan(h^2)}{h}

= \lim_{h \to 0} |h|\tan(h^2)=0

So,

p'(0)=0

exists.

Thus,

p(x)

is differentiable at

x=0

Answer:

Both I and II are correct.

Option C: Both I and II.

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