Challenging +1.2 This is a multi-step calculus problem requiring differentiation to find the relationship between a and b using the minimum condition, then integration with the area constraint to solve for the constants, and finally substitution to find the y-coordinate. While it involves several techniques (differentiation, stationary points, definite integration), each step follows standard A-level procedures without requiring novel insight. The algebraic manipulation is moderately involved but straightforward for Further Maths students.
The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\).
\includegraphics{figure_1}
The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\).
Given that the area of the shaded region is 48 units\(^2\), determine the \(y\)-coordinate of \(P\).
[7]
The diagram shows the curve with equation $y = 5x^4 + ax^3 + bx$, where $a$ and $b$ are integers. The curve has a minimum at the point $P$ where $x = 2$.
\includegraphics{figure_1}
The shaded region is enclosed by the curve, the $x$-axis and the line $x = 2$.
Given that the area of the shaded region is 48 units$^2$, determine the $y$-coordinate of $P$.
[7]
\hfill \mbox{\textit{SPS SPS FM 2025 Q1 [7]}}