A curve $y=f(x)$ is such that $f'(x)=ax^2-6x+b$, where $a$ and $b$ are constants.

Question

A curve $y=f(x)$ is such that $f'(x)=ax^2-6x+b$, where $a$ and $b$ are constants. Given that the curve is always increasing, what conditions must apply to $a$ and $b$?[3] It is now given that $a=5$. The curve intersects the $y$-axis at $(0,4)$ and passes through the points $(-3,-80)$ and $(3,k)$. Find the value of $k$.[5] This question is not related to part (a). It is given that $y=g(x)$ is a solution of the equation $e^{-2x}\left(\dfrac{d^2y}{dx^2}-2\dfrac{dy}{dx}+y\right)=-e^k$, where $k$ is a constant. The point $(1,0)$ is a stationary point on the curve $y=g(x)$. Find the nature of this stationary point.[3]

Accepted Answer

(a)(i) $a>0$ and $ab\ge9$. (a)(ii) $k=34$. (b) $(1,0)$ is a maximum point.

(A Math) 2025 CCHY PRELIMS P2 Q11

(A Math) 2025 CCHY PRELIMS P2 Q11

Solution:

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