Question

For the function f(x) = 2e^5x + 10, which of the following is the most appropriate option.

• A. The minimum value of f is 10
• B. f has no maximum possible value.
• C. f has no minimum possible value.
• D. f has neither maximum nor minimum possible value.

Answer 1

The Correct Option is D

To analyze the function \(f(x) = 2e^{5x} + 10\), we need to understand the behavior of the exponential function \(e^{5x}\). The function \(e^x\) is known to increase rapidly as \(x\) becomes larger and tends towards zero as \(x\) becomes very negative.

Step-by-step Analysis:

1. Minimum Value Analysis: Since \(e^{5x}\) is always positive, the expression \(2e^{5x}\) will always be greater than zero. Adding 10 to it \( (2e^{5x} + 10)\) implies that the minimum value of \(f(x)\) will be 10, which is simply reached as \(x\) approaches \(-\infty\). However, \(f(x)\) approaches 10 but never actually reaches a value less than 10, negating any minimum value finitely.

2. Maximum Value Analysis: Because \(2e^{5x}\) grows large without bound as \(x\) increases, \(f(x) = 2e^{5x} + 10\) similarly grows without limit. This shows that \(f(x)\) does not have a maximum finite value.

Combining these observations, \(f(x)\) has neither a strict minimum nor maximum finite value. Therefore, the most appropriate option is: f has neither maximum nor minimum possible value.