For \(y = -x^{3} + 6x^{2} + 2x - 5\), the maximum value of the rate of change of y with respect to x is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 14
The rate of change of y with respect to x is the derivative, so the function to maximise is
\( \displaystyle g(x) = \frac{dy}{dx} = -3x^{2} + 12x + 2 \)
Setting its derivative equal to zero:
\( g'(x) = -6x + 12 = 0,\ x = 2 \)
Since \(g''(x) = -6 \lt 0\), this is a maximum of g:
\( g(2) = -12 + 24 + 2 = 14 \)
Why the other options are incorrect
Option (1) 15 — Incorrect. This is the value of y itself at x = 2: \(-8 + 24 + 4 - 5 = 15\), not the value of \(\frac{dy}{dx}\).
Option (2) 2 — Incorrect. This is the point x = 2 where the maximum occurs, not the maximum value.
Option (4) 12 — Incorrect. g(2) = −12 + 24 + 2 = 14, not 12.