Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

The minimum value of \(a \tan ^{2} x+b \cot ^{2} x\) equals the maximum value of \(a \sin ^{2} \theta+b \cos ^{2} \theta\) where \(a>b>0\), when (A) \(a=b\) (B) \(a=2 b\) (C) \(a=3 b\) (D) \(a=4 b\)

Step-by-Step Solution

Answer

The answer is (D) \(a=4b\).

Step 1: Expression for Minimum Value

To find the minimum value of the expression \(a \tan^2 x + b \cot^2 x\), we express it as \(a \frac{\sin^2 x}{\cos^2 x} + b \frac{\cos^2 x}{\sin^2 x}\). Let \(t = \tan^2 x\). Then the expression becomes \(a t + \frac{b}{t}\).

Step 2: Using AM-GM Inequality

By the AM-GM inequality, \(a t + \frac{b}{t} \geq 2 \sqrt{a \cdot \frac{b}{t} \cdot t} = 2 \sqrt{ab}\). The equality holds when \(a t = \frac{b}{t}\), giving \(t = \sqrt{\frac{b}{a}}\). Thus, the minimum value is \(2 \sqrt{ab}\).

Step 3: Expression for Maximum Value

To find the maximum value of \(a \sin^2 \theta + b \cos^2 \theta\), we express it as \(a \sin^2 \theta + b (1 - \sin^2 \theta) = (a-b) \sin^2 \theta + b\). This is a quadratic in \(\sin^2 \theta\).

Step 4: Maximizing the Quadratic Expression

The expression \((a-b) \sin^2 \theta + b\) is maximized when \(\sin^2 \theta = 1\) if \(a-b > 0\). Thus, the maximum value is \(a\).

Step 5: Equating Minimum and Maximum Values

We set the minimum value \(2 \sqrt{ab}\) equal to the maximum value \(a\). Solving the equation \(a = 2 \sqrt{ab}\), square both sides to get \(a^2 = 4ab\) leading to \(a = 4b\).

Step 6: Conclusion

From the calculations, the correct answer is choice \(a=4b\) that satisfies the requirement of the minimum value of the given expression equaling the maximum value.

Key Concepts

Trigonometric Identities

In trigonometry, identities play a crucial role in simplifying and solving equations. A **trigonometric identity** is an equation involving trigonometric functions that is true for all angles. These identities help us transform complex expressions into more manageable forms.

For example, the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \), is fundamental when dealing with expressions like \( a \sin^2 \theta + b \cos^2 \theta \). By substituting trigonometric identities, we can rewrite expressions to make calculations easier or extract maximum or minimum values.

Another important identity is \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \). This allows expressions involving tangent and cotangent, like \( a \tan^2 x + b \cot^2 x \), to be written in a way that's more convenient for application of tools like the AM-GM inequality. Identifying and using these identities effectively is key to solving many trigonometric problems in mathematics.

AM-GM Inequality

The **Arithmetic Mean-Geometric Mean (AM-GM) Inequality** is a useful tool in optimization problems. It states that for any non-negative real numbers \( x \) and \( y \), the arithmetic mean is always greater than or equal to the geometric mean:

• AM: \( \frac{x + y}{2} \)
• GM: \( \sqrt{xy} \)

Thus, \( \frac{x + y}{2} \geq \sqrt{xy} \), with equality if and only if \( x = y \).

In the problem, we utilize this inequality to find the minimum of \( a t + \frac{b}{t} \) where \( t = \tan^2 x \). Applying AM-GM to the terms gives \( a t + \frac{b}{t} \geq 2\sqrt{ab} \), showing that the minimum value occurs when \( t = \sqrt{\frac{b}{a}} \). The AM-GM inequality is particularly handy in proving minimum/maximum values where variables are constrained by a product or sum.

Quadratic Optimization

**Quadratic optimization** involves finding the extrema (minimum or maximum values) of quadratic functions. Such functions can be of the form \( ax^2 + bx + c \). Optimization requires analyzing the function's behavior, usually its vertex, which gives the function's minimum or maximum value.

In this exercise, to find the maximum of \( a \sin^2 \theta + b \cos^2 \theta \), we rearrange it as \( (a - b) \sin^2 \theta + b \). This peaks when \( \sin^2 \theta = 1 \), if \( a > b \), yielding a maximum value of \( a \).

Quadratic optimization involves using calculus, completing the square, or leveraging symmetrical properties to efficiently/effectively find these extreme points. Understanding this process is crucial in many applied mathematics fields, including physics and economics, where optimizing outcomes is often required.