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Consider the curve $\Gamma$ : $y = a(x - 1)^3 + b(x - 1)^2 + cx + 2$, where $a$ , $b$, and $c$ are constants. It is given that the curve has a maximum point of $\Gamma$ at $x = 1$.

(a) Find $c$.

(b) It is given that the y-intercept of $\Gamma$ is $1$ and $ab = 6$.

(i) Someone claims that there are two possible values of $a$. Do you agree? Explain your answer.

(ii) Find the other extreme point(s) of $\Gamma$.

Worked Solution

In bi), as $x = 1$ is a maximum point of $\Gamma$, therefore $\frac{d^2y}{dx^2}\Big|_{x=1} < 0$.

Solving the inequality, we have $b < 0$.