For the function \(f(x) = 2e^{5x} + 10\), which of the following is the most appropriate option.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → f has neither maximum nor minimum possible value.
The derivative is
\(f'(x) = 10e^{5x}\)
Since \(e^{5x} \gt 0\) for every real x, \(f'(x) \gt 0\) everywhere. So f is strictly increasing on R and has no critical point.
As \(x \to \infty\), \(f(x) \to \infty\): there is no largest value.
As \(x \to -\infty\), \(e^{5x} \to 0\), so \(f(x) \to 10\), but \(f(x) \gt 10\) for every x.
The value 10 is approached but never attained, so f has no smallest value either. Hence f has neither a maximum nor a minimum value.
Why the other options are incorrect
Option (1) The minimum value of f is 10 — Incorrect. \(f(x) = 10\) would need \(e^{5x} = 0\), which never happens; 10 is only a lower bound.
Option (2) f has no maximum possible value. — Incorrect. True, but it tells only half the story; f has no minimum value either, so it is not the most appropriate option.
Option (3) f has no minimum possible value. — Incorrect. True, but incomplete in the same way; f has no maximum value either.