Question

The minimum value of \(e^{2x}+e^{-2x}\) is .......

• A. \(-2\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

Use AM-GM or set the derivative to zero.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Let \(y=e^{2x}+e^{-2x}\). Differentiate to find the critical point.

Step 2: First Derivative
\(y'=2e^{2x}-2e^{-2x}=0\Rightarrow e^{4x}=1\Rightarrow x=0\).

Step 3: Second Derivative
\(y''=4e^{2x}+4e^{-2x}\), which at \(x=0\) equals 8, a positive value, so \(x=0\) is a minimum.

Step 4: Minimum Value
\(y(0)=1+1=2\), option (D).

Final Answer:
The minimum value is 2. \[ \boxed{\text{(D) }2} \]