Problem 40 - Calculus for JEE ADVANCED

Question

\(\begin{aligned} y=f(x) &=\cos ^{2} x+\sin ^{4} x \\ &=\cos ^{2} x+\sin ^{2} x\left(1-\cos ^{2} x\right) \\ &=\cos ^{2} x+\sin ^{2} x-\sin ^{2} x \cos ^{2} x \\ &=1-\sin ^{2} x \cos ^{2} x \\ &=1-\frac{1}{4} \sin ^{2} 2 x \\\ \therefore \quad & \frac{3}{4} \leq f(x) \leq 1 \\ \therefore \quad f(x) & \in[3 / 4,1] \end{aligned}\)

Step-by-Step Solution

Answer

The range of \( f(x) \) is \([\frac{3}{4}, 1]\).

Step 1: Simplify Trigonometric Expression

Given the expression \( y = \cos^2 x + \sin^4 x \), first, we'll rewrite \( \sin^4 x \) as \( \sin^2 x (\sin^2 x) \). Substitute \( \sin^2 x = 1 - \cos^2 x \), resulting in \( y = \cos^2 x + \sin^2 x (1 - \cos^2 x) \).

Step 2: Distribute and Combine Terms

Simplify the expression from step 1: \( y = \cos^2 x + \sin^2 x - \sin^2 x \cos^2 x \). Combine terms to yield \( y = 1 - \sin^2 x \cos^2 x \).

Step 3: Use Double Angle Identity

Recall and apply the double angle identity \( \sin 2x = 2 \sin x \cos x \). Hence, \( \sin^2 2x = 4 \sin^2 x \cos^2 x \). Substitute in the expression to get \( y = 1 - \frac{1}{4} \sin^2 2x \).

Step 4: Determine Range of Function

Since \( \sin^2 2x \) ranges between 0 and 1 for all \( x \), \( \frac{1}{4}\sin^2 2x \) ranges between 0 and \( \frac{1}{4} \). This results in the range of \( y = 1 - \frac{1}{4}\sin^2 2x \) being \( \frac{3}{4} \leq y \leq 1 \).

Step 5: Conclude the Range of \( f(x) \)

From step 4, we determine that the range of \( f(x) \) is \([\frac{3}{4}, 1]\).

Key Concepts

Function Range

The function range is a crucial concept in mathematics that tells us the possible output values a function can produce. For any function, the range is the set of y-values that we can obtain by plugging all possible x-values from the domain into the function. For the function \( f(x) = \cos^2 x + \sin^4 x \), we carefully analyze and transform it to find its range.
This function can be rewritten using identities we explore in the subsequent sections. As we simplify to \( y = 1 - \frac{1}{4} \sin^2 2x \), we need to understand the behavior of \( \sin^2 2x \).

• The expression \( \sin^2 2x \) has the possible value range \([0, 1]\).
• Thus, \( \frac{1}{4} \sin^2 2x \) ranges between \(0\) and \(\frac{1}{4}\).
• This alters our function to have an output range of \( [\frac{3}{4}, 1] \), as \( y = 1 - \) any value from \([0, \frac{1}{4}]\).

Understanding range tells us all potential y-values of \( f(x) \) and ensures we're considering the function's complete behavior.

Double Angle Identity

The double angle identity is one of the fundamental identities in trigonometry and it greatly simplifies certain trigonometric expressions. The identity for sine is \( \sin 2x = 2 \sin x \cos x \). This provides a link between single angle functions and double angles.
For our function \( y = 1 - \sin^2 x \cos^2 x \), the utilization of this identity paves the way for further simplification.

• Using the identity, \( \sin^2 2x = (2 \sin x \cos x)^2 = 4 \sin^2 x \cos^2 x \).
• By rearranging, we express \( \sin^2 x \cos^2 x \) as \( \frac{1}{4} \sin^2 2x \).

Substituting this into the expression for \( y \) gives \( y = 1 - \frac{1}{4} \sin^2 2x \). This step is crucial as it allows us to delve deeper into understanding the output range based on trigonometric properties of the sine function.

Trigonometric Simplification

Simplifying trigonometric expressions involves using identities and algebra to make expressions more manageable. For \( y = \cos^2 x + \sin^4 x \), the goal is to rewrite it in simpler or more revealing forms. This often utilizes identities like \( \sin^2 x = 1 - \cos^2 x \) and products like \( \sin^2 x \cos^2 x \).
Breaking down our function step by step:

• Start with rewriting \( \sin^4 x \) as \( \sin^2 x (1 - \cos^2 x) \), making it easier to combine like terms.
• Next, distribute and combine: \( y = \cos^2 x + \sin^2 x - \sin^2 x \cos^2 x \).
• This transformation leads to \( y = 1 - \sin^2 x \cos^2 x \).

Such simplification uncovers deeper properties of the function, which then allow us to apply trigonometric identities effectively. Trigonometric simplification is an essential skill, especially when dealing with complex functions. It reduces complexity, making more strategic plays like double angle identities applicable.