The maximum and minimum values of f(x)=3 sin^2 x+4 cos^2 x is (a) {-4,-3} (b) {7,3} (c) {4,-3} (

step 1 solving the problem f x equal to three sine square x plus four times of cos square x we'll use t

The maximum and minimum values of f(x)=3 sin^2 x+4 cos^2 x...

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Key Concepts

Pythagorean Identity

The Pythagorean identity, sin²x + cos²x = 1, is a fundamental relation in trigonometry that allows one to express one trigonometric squared term in terms of the other. This identity is crucial for simplifying expressions and converting functions that combine sin²x and cos²x into a single variable function, making it easier to analyze their behavior.

Range of Squared Trigonometric Functions

Since sin²x (and similarly cos²x) represents the square of a sine or cosine function, its value is always non-negative and lies between 0 and 1. Understanding this range is important when determining the maximum and minimum values of functions involving these terms, as it confines the potential outputs and simplifies the optimization process.

Function Transformation for Extremum Analysis

Transforming a trigonometric function into a simpler form by substituting one trigonometric squared term using the Pythagorean identity aids in isolating the variable of interest. In many cases, this transformation produces a function whose extremum over a known interval (like [0, 1] for sin²x or cos²x) can be found using basic algebraic techniques, thereby finding the minimum and maximum values of the original function.

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