UPSC NDA I 2026 Mathematics Paper

Question Numbers: 85-86

For the next two (02) items that follow:

Let S and T be the sets where

f(x)=\frac{x^{3}}{3}-\frac{5 x^{2}}{2}+6 x+7

decreases and increases respectively.

Question : 86 of 120

Marks: +1, -0

What is S equal to ?

Solution:

Concept:

A function decreases where its derivative is negative.

Explanation:

f(x)=\frac{x^{3}}{3}-\frac{5x^{2}}{2}+6x+7

f'(x)=x^{2}-5x+6=(x-2)(x-3)

For

S

f(x)

decreases, so

f'(x)<0

(x-2)(x-3)<0

This inequality holds for

2 < x<3

Therefore,

S=(2,3)

Answer:

S=(2,3)

, i.e., option C.

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