Question

Let \( A \) and \( G \) denote the arithmetic mean and geometric mean of positive real numbers \( 5^x \) and \( 5^{1 - x} \). Then the minimum value of the expression \[ 5^x + 5^{1 - x}, \text{ where } x \in \mathbb{R}, \text{ is:} \]

• A. \( 2\sqrt{5} \)
• B. 0
• C. 1
• D. \( \sqrt{5} \)

Hint:

The AM-GM inequality states that the arithmetic mean is always greater than or equal to the geometric mean, and equality holds when all the terms are equal.

Answer 1

The Correct Option is A

Step 1: Arithmetic and geometric means.
The arithmetic mean \( A \) of two numbers \( a \) and \( b \) is given by:
\[ A = \frac{a + b}{2} \]
The geometric mean \( G \) of two numbers \( a \) and \( b \) is given by:
\[ G = \sqrt{ab} \]
Here, \( a = 5^x \) and \( b = 5^{1 - x} \).

Step 2: Use the given formulas.
The arithmetic mean \( A \) of \( 5^x \) and \( 5^{1 - x} \) is:
\[ A = \frac{5^x + 5^{1 - x}}{2} \]
The geometric mean \( G \) of \( 5^x \) and \( 5^{1 - x} \) is:
\[ G = \sqrt{5^x \cdot 5^{1 - x}} = \sqrt{5^1} = \sqrt{5} \]

Step 3: Use the AM-GM inequality.
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality: \[ A \geq G \]
Substitute the values of \( A \) and \( G \) into this inequality: \[ \frac{5^x + 5^{1 - x}}{2} \geq \sqrt{5} \]
Multiply both sides by 2: \[ 5^x + 5^{1 - x} \geq 2\sqrt{5} \]

Step 4: Check when equality holds.
Equality in the AM-GM inequality holds when \( 5^x = 5^{1 - x} \), which means:
\[ x = 1 - x \]
Solving for \( x \): \[ 2x = 1 \quad \Rightarrow \quad x = \frac{1}{2} \]

Step 5: Find the value of the expression at \( x = \frac{1}{2} \).
Substitute \( x = \frac{1}{2} \) into the expression \( 5^x + 5^{1 - x} \):
\[ 5^{1/2} + 5^{1 - 1/2} = \sqrt{5} + \sqrt{5} = 2\sqrt{5} \]

Step 6: Conclusion.
Thus, the minimum value of the expression is \( 2\sqrt{5} \), which corresponds to option (A).