The maximum value of \(f(x) = 5 + 8x - 2x^2\), \(x \in R\), is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 13
At a maximum the derivative is zero:
\(f'(x) = 8 - 4x\)
\(8 - 4x = 0\)
\(x = 2\)
Since \(f''(x) = -4 \lt 0\), the function has its maximum at x = 2.
\(f(2) = 5 + 16 - 8\)
\(f(2) = 13\)
Why the other options are incorrect
Option (2) 2 — Incorrect. This is the point x = 2 where the maximum occurs, not the maximum value.
Option (3) 5 — Incorrect. This is f(0); the value at x = 2 is larger.
Option (4) 21 — Incorrect. This is 5 + 16, leaving out the term \(-2x^2 = -8\).