Problem 77 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Solution of \(2^{x}+2^{|x|} \geq 2 \sqrt{2}\) is (A) \(\left(-\infty, \log _{2}(\sqrt{2}+1)\right.\) (B) \((0,8)\) (C) \(\left(\frac{1}{2}, \log _{2}(\sqrt{2}-1)\right)\) (D) \(\left(-\infty, \log _{2}(\sqrt{2}-1)\right] \cup\left[\frac{1}{2}, \infty\right)\)

Step-by-Step Solution

Answer

The correct solution is (D): \((-\infty, \log_2(\sqrt{2} - 1)] \cup [1/2, \infty)\).

Step 1: Understand the Problem

We need to find the values of \(x\) that satisfy the inequality \(2^{x} + 2^{|x|} \geq 2\sqrt{2}\). This involves understanding how both terms vary with \(x\) by evaluating the expression in different cases based on the value of \(x\).

Step 2: Consider Case 1: \(x \geq 0\)

When \(x \geq 0\), \(|x| = x\), so the inequality becomes \(2^x + 2^x \geq 2\sqrt{2}\), or \(2 \cdot 2^x \geq 2\sqrt{2}\). Simplifying, we get \(2^{x} \geq \sqrt{2}\) which implies \(2^{x} \geq 2^{1/2}\). Therefore, \(x \geq 1/2\).

Step 3: Consider Case 2: \(x < 0\)

When \(x < 0\), \(|x| = -x\). Thus, the inequality becomes \(2^{x} + 2^{-x} \geq 2\sqrt{2}\). To solve it, set \(y = 2^x\), then the inequality is \(y + \frac{1}{y} \geq 2\sqrt{2}\). Testing with critical points around symmetric geometrical points derived from AM-GM inequality and specific values such as \(y = 1\), we find that the inequality holds for \( y \leq \sqrt{2} - 1 \), which translates back to \(x \leq \log_2(\sqrt{2} - 1)\).

Step 4: Combine Results

From Case 1, we have \(x \geq 1/2\). From Case 2, we determined \(x \leq \log_2(\sqrt{2} - 1)\). Combine these solutions to find the intervals where the original inequality holds.

Step 5: Conclusion

The combined solution is \(x \leq \log_2(\sqrt{2} - 1)\) or \(x \geq 1/2\), which corresponds to choice (D): \((-\infty, \log_2(\sqrt{2} - 1)] \cup [1/2, \infty)\).

Key Concepts

Inequality Solving

Inequalities are mathematical expressions involving symbols like ">", "<", ">=", or "<=" that describe a range of values rather than a single value. Understanding inequalities is crucial in solving problems where multiple solutions can exist. Let's break down how to solve an inequality step by step.

First, always start by simplifying both sides of the inequality if possible. Look for like terms or common factors. This is similar to simplifying equations.

After simplifying, determine the conditions under which the inequality holds true. It may help to split the problem into different cases. In the given example, the problem involves splitting into separate cases where the absolute value term changes behavior based on whether the variable inside is negative or non-negative.

• Case Analysis: Consider scenarios such as when the variable is positive or negative.
• Comparison to Constants: Rearrange the inequality to directly compare your expression to known numbers or constant expressions.

Finally, compile the results from each case analysis. The solution may be a combination of intervals. Like other mathematical problems, check the boundary conditions to ensure they satisfy the original inequality.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. These functions can grow very quickly and are represented by expressions such as \(2^x\). Understanding exponential functions is crucial when working with inequalities involving terms like \(2^x\).

Here are some key properties of exponential functions that are helpful while solving inequalities:

• Base Greater Than 1: If the base of the exponential is greater than 1, the function will increase as the exponent increases.
• Equality and Inequality: Use the property that if \(a^b = a^c\), then \(b = c\), provided \(a > 0\) and \(a eq 1\). This helps in transforming inequalities.

When dealing with inequalities, you often need to compare exponential expressions with constants or other functions. This can involve logarithms to help solve for the variable within the exponent.

In our exercise, solving \(2^x \geq \, \sqrt{2}\) turned into assessing when\(2^x\) meets or exceeds a particular value, here \(2^{1/2}\), using knowledge of exponential growth.

Absolute Value

Absolute value gives the non-negative magnitude of a number, effectively ignoring its sign. In inequalities, absolute value impacts how you interpret and solve for variables. The notation \(|x|\) implies a piecewise function: \(x\) when \(x \geq 0\) and \(-x\) when \(x < 0\).

To solve inequalities with absolute values, consider separating them into different cases:

• Non-negative Values: Consider the direct form where the absolute value can be removed without altering the sign.
• Negative Values: Formulate the inequality assuming the value inside the absolute sign is negative, necessitating a change in the sign.

These cases allow the inequality to be rewritten so that the absolute value no longer "hides" any negative sign. This makes it easier to directly apply algebraic methods to solve the problem.

As seen in the original exercise, different scenarios based on \(x\) values—positive or negative—were analyzed separately. This piecewise approach simplifies understanding how \(|x|\) affects the rest of the equation.