A curve has the equation $y = \frac{(x + a)(2x + b)}{x + 2}$ where $a$ and $b$ are intege

Question

A curve has the equation $y = \frac{(x + a)(2x + b)}{x + 2}$ where $a$ and $b$ are integers. Given that the curve passes through $(2, 0)$ and that the tangent to the curve at $x = 0$ is parallel to $x + y + 2 = 0$, find the values of $a$ and $b$. Hence find the coordinates of the points on the curve where the gradient of the curve is $-1$, and the range of values of $x$ where the gradient of the curve is negative.

Accepted Answer

(i) $a=-2,\ b=1$ (ii) $(0,-1)$ and $(-4,-21)$ (iii) $-2-\sqrt6<x<-2$ or $-2<x<-2+\sqrt6$

Differentiation - Tangents and Normals at a Point

Differentiation - Tangents and Normals at a Point

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