Problem 15 - A Complete Resource Book in Mathematics for JEE Main

Question

The minimum value of \(8^{\sin x^{\prime} 8}+8^{\cos x^{\prime} 8}\) is (A) \(2^{\frac{1}{3-\sqrt{2} / \sqrt{2}}}\) (B) \(2^{\frac{3+\sqrt{2}}{\sqrt{2}}}\) (C) \(2^{\frac{1}{3+\sqrt{2} / \sqrt{2}}}\) (D) \(2^{\frac{3-\sqrt{2}}{\sqrt{2}}}\)

Step-by-Step Solution

Answer

The minimum value is \(2^{\frac{5}{2}}\).

Step 1: Understand the Expression

We are given the expression \(8^{\sin x}+8^{\cos x}\). We need to find its minimum value.

Step 2: Express in Common Base

Rewrite 8 as a power of 2: \(8 = 2^3\). Thus, the expression becomes \((2^3)^{\sin x} + (2^3)^{\cos x}\). This simplifies to \(2^{3\sin x} + 2^{3\cos x}\).

Step 3: Introduce a Function

Let's consider the function \(f(a) = 2^a + 2^{3-a}\), where \(a = 3\sin x\). Since \(\sin^2 x + \cos^2 x = 1\), it implies \(0 \leq a \leq 3\).

Step 4: Apply AM-GM Inequality

To find the minimum of \(f(a) = 2^a + 2^{3-a}\), apply the Arithmetic Mean-Geometric Mean (AM-GM) Inequality: \(\frac{2^a + 2^{3-a}}{2} \geq \sqrt{2^a \cdot 2^{3-a}} = \sqrt{2^3} = 2^{3/2}\). Simplifying gives \(2^a + 2^{3-a} \geq 2^{3/2+1} = 2^{2.5}\).

Step 5: Determine Exact Minimum

The inequality achieves equality when \(2^a = 2^{3-a}\). Solving for \(a\), we get \(a = \frac{3}{2}\). Thus, \(\sin x = \cos x = \frac{\sqrt{2}}{2}\), meaning \(x = \frac{\pi}{4} + k\pi, k \in \mathbb{Z}\).

Step 6: Minimum Value Calculation

Substituting \(a = \frac{3}{2}\) gives \(2^a + 2^{3-a} = 2^{3/2} + 2^{3/2} = 2 \cdot 2^{3/2} = 2^{2.5}\). This is \(2^{5/2}\), which can be written as \(2^{\frac{5}{2}}\).

Key Concepts

AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) Inequality is a powerful tool used in mathematics to help find the minimum or maximum value of certain expressions. It's grounded in the principle that the average (or mean) of a set of numbers is at least as large as the geometric mean of those numbers.

For two non-negative numbers, say \(a\) and \(b\), the inequality is expressed as:

• \(\frac{a + b}{2} \geq \sqrt{ab}\)

This means that the arithmetic mean, \(\frac{a+b}{2}\), is greater than or equal to the geometric mean, \(\sqrt{ab}\).

In the exercise, we applied the AM-GM inequality to the expression \(f(a) = 2^a + 2^{3-a}\). By using this inequality, we showed:

• \(\frac{2^a + 2^{3-a}}{2} \geq \sqrt{2^a \cdot 2^{3-a}} = \sqrt{2^3} = 2^{3/2}\).

This implies that the minimum value of \(2^a + 2^{3-a}\) is achieved when the two terms are equal, i.e., when \(a = \frac{3}{2}\).

Understanding how and when to apply this inequality can greatly simplify your problem-solving process with expressions that can be complex at first glance.

Exponents and Powers

Exponents and powers are fundamental concepts in mathematics that help in expressing repeated multiplication. Understanding these can simplify complex calculations significantly.
A power is made up of a base and an exponent, written as \(b^n\), where \(b\) is the base and \(n\) is the exponent. This represents \(b\) multiplied by itself \(n\) times.
In our problem, recognizing that \(8 = 2^3\) allowed us to rewrite the expression \(8^{\sin x} + 8^{\cos x}\) in terms of a common base (2). This conversion is expressed as:

• \(8^{\sin x} = (2^3)^{\sin x} = 2^{3\sin x}\)
• \(8^{\cos x} = (2^3)^{\cos x} = 2^{3\cos x}\)

Thus, the problem simplifies to \(2^{3\sin x} + 2^{3\cos x}\).
Dealing with exponents properly is key in manipulating and solving expressions effectively, especially when integrating bases to find optimal solutions.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They are used to simplify and solve trigonometric expressions.
One of the most fundamental identities is the Pythagorean identity:

• \(\sin^2 x + \cos^2 x = 1\)

This identity tells us that for any angle \(x\), the square of the sine plus the square of the cosine equals 1. In our exercise, this was critical in constraining the range for \(a = 3\sin x\) as:

• Since \(\sin x\) and \(\cos x\) are complementary parts of a ratio formed by 1, \(0 \leq a \leq 3\).

This condition was pivotal in determining the possible values for \(a\) which influence the potential values of \(2^a\) and \(2^{3-a}\).
Also, knowing that \(\sin x = \cos x = \frac{\sqrt{2}}{2}\) implies \(x = \frac{\pi}{4} + k\pi\), where \(k\) is an integer, helped us find the point at which the minimum value was achieved. Understanding these identities allows us to find such solutions efficiently.