A-Level Mathematics May/June 2025 Q7(b): It is given that the maximum point of the curve has y-coordinate 27. Find the equation…

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  From part (a), the derivative of the curve is $\frac{dy}{dx} = 3x^2 + 10x - 8$. The stationary points occur at $x = -4$ and $x = \frac{2}{3}$. The maximum point is at $x = -4$.

  It is given that the y-coordinate of the maximum point is 27. Therefore, the curve passes through the point $(-4, 27)$.

  To find the equation of the curve, we integrate the derivative $\frac{dy}{dx}$: $y = \int (3x^2 + 10x - 8) \,dx$ $y = \frac{3x^3}{3} + \frac{10x^2}{2} - 8x + c$ $y = x^3 + 5x^2 - 8x + c$

  Now, substitute the coordinates of the maximum point $(-4, 27)$ into the equation to find the constant of integration, $c$. $27 = (-4)^3 + 5(-4)^2 - 8(-4) + c$ $27 = -64 + 5(16) + 32 + c$ $27 = -64 + 80 + 32 + c$ $27 = 48 + c$ $c = 27 - 48$ $c = -21$

  Therefore, the equation of the curve is: $y = x^3 + 5x^2 - 8x - 21$

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  How the marks are awarded

  • B1 — Identifying that the maximum point occurs at x = -4. This is stated at the start and used in the substitution.
  • B1 — Correctly integrating the expression for dy/dx to get y = x³ + 5x² - 8x + c. All three terms must be correct and the constant of integration must be included.
  • M1 — Substituting the known point (their x-value for the maximum, -4, and the given y-value, 27) into their integrated cubic equation to find the constant c.
  • A1 — Correctly calculating c = -21 and stating the final equation of the curve as y = x³ + 5x² - 8x - 21.

  Common mistakes

  • Using the x-coordinate of the minimum point (x = 2/3) instead of the maximum point (x = -4) to find the constant c.
  • Forgetting to include the constant of integration, '+c', after integrating, which makes it impossible to find the specific equation.
  • Making arithmetic errors when substituting x = -4, particularly with negative signs, for example calculating (-4)³ as 64 or 5(-4)² as -80.
  • Incorrectly integrating one or more terms, for example integrating -8 to -8 instead of -8x.

  Examiner tip: To find the specific equation of a curve from its derivative, you must first integrate to find the general equation with a constant '+c', then substitute the coordinates of a known point on the curve to solve for c.

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