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UPSC CDS 1 2023 Math Paper

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Question : 49 of 100

Marks: +1, -0

What is the minimum value of 6 - 4 sin θ, 0 ≤ θ ≤

\frac{\pi }{2}

1
2
4
6

Solution:

Concept:

The expression is minimized when the subtracted term

4 \sin \theta

is largest.

Explanation:

The function is

f(\theta) = 6 - 4 \sin \theta

.

Since

\theta

lies in

[0, \frac{\pi}{2}]

,

\sin \theta

ranges from

0

to

1

.

The maximum of

\sin \theta

in this interval is

1

.

To minimize

f(\theta)

, we use the largest value of

4 \sin \theta

, which is

4 \times 1 = 4

.

Thus

f(\theta)_{\text{min}} = 6 - 4 = 2

.

Answer:

2

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