If \(C(x) = 15x^3 - 3\sqrt{5}x + \sqrt{7}\) represents the total cost of producing x items, the slope of the marginal cost function at x = 1 is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 90
The marginal cost is the derivative of the total cost:
\( \displaystyle MC = \frac{dC}{dx} = 45x^2 - 3\sqrt{5} \)
The slope of the marginal cost function is the derivative of MC:
\( \displaystyle \frac{d(MC)}{dx} = 90x \)
At x = 1:
\(90(1) = 90\)
Why the other options are incorrect
Option (1) \(45 - 3\sqrt{5}\) — Incorrect. This is the marginal cost itself at x = 1, not the slope of the marginal cost function.
Option (2) \(90 + 3\sqrt{5}\) — Incorrect. The constant −3√5 in MC becomes 0 on differentiating, so nothing is added to 90.
Option (3) \(\displaystyle\frac{15x^4}{4} - \frac{3\sqrt{5}}{2} + \sqrt{7}x + C\) — Incorrect. This comes from integrating C(x), not differentiating it, and it is not even a number.