An extreme value of 4 sin^2 x+3 cos^2 x-24 sin(x)/(2) -24 cos(x)/(2), where 0 ≤x ≤(π)/(2), is (A

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An extreme value of 4 sin^2 x+3 cos^2 x-24 sin(x)/(2) -24...

An extreme value of $4 \sin ^{2} x+3 \cos ^{2} x-24 \sin \frac{x}{2}$
$-24 \cos \frac{\mathrm{x}}{2}$, where $0 \leq \mathrm{x} \leq \frac{\pi}{2}$, is
(A) $4+\sqrt{2}$
(B) $4(1-6 \sqrt{2})$
(C) $-21$
(D) 4

Key Concepts

Extreme Value Problems

This concept involves finding the maximum or minimum values of a function, usually by applying techniques from calculus such as taking the derivative and setting it equal to zero, or by analyzing the endpoints of the interval if the function is continuous over a closed interval.

Trigonometric Identities

Understanding trigonometric identities, such as the Pythagorean identity, half-angle formulas, and double-angle formulas, is crucial. These identities enable one to simplify and transform trigonometric expressions into more workable forms, which is often a necessary step when solving optimization problems that involve trigonometric functions.

Optimization with Domain Constraints

This relates to solving optimization problems where the variable is limited to a specific interval. It is important to analyze both the interior points (by finding critical points where derivatives vanish) and the boundary values (the endpoints of the interval) to determine the absolute maximum or minimum values of the function.

Function Transformation and Substitution

Often, complex trigonometric expressions can be simplified by substitution or rewriting certain parts of the function in equivalent forms. By expressing functions involving angles like x in terms of half-angles or other transformed variables, one can simplify the overall problem and make the process of finding extreme values more straightforward.

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