If \(y = 2x^3 - 5x^2 + 7x - 1\), then the value of \(\displaystyle\frac{d^2y}{dx^2}\) at x = 2 is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 14
Differentiating the polynomial term by term with the power rule:
\( \displaystyle \frac{dy}{dx} = 6x^2 - 10x + 7 \)
Differentiating once more gives the second derivative:
\( \displaystyle \frac{d^2y}{dx^2} = 12x - 10 \)
At x = 2:
\( \displaystyle \frac{d^2y}{dx^2} = 24 - 10 = 14 \)
Why the other options are incorrect
Option (1) 11 — Incorrect. This is the first derivative at x = 2: 24 − 20 + 7 = 11.
Option (2) 24 — Incorrect. Only the term 12x has been used; the constant −10 has been dropped.
Option (4) 2 — Incorrect. This is the second derivative at x = 1, not at x = 2.