The curve $C$ has equation $y=\frac{{{x}^{2}}+3x+b}{x+a}$ $(x
e -a)$, where $a$ and $b$ are constan

Question

The curve $C$ has equation $y=\frac{{{x}^{2}}+3x+b}{x+a}$ $(x
e -a)$, where $a$ and $b$ are constants. It is given that the line $y=x+4$is an asymptote of $C$, and that $C$ has a stationary point at $x=2$. Show that $a=-1$, and find the value of $b$. [6] Using the same values of $a$ and $b$, Show that $C$ has exactly two stationary points, and find the coordinates of each stationary point, [3] Draw a sketch of $C$, [2] Deduce that, for all positive values of $k$, the equation ${{x}^{3}}+3{{x}^{2}}+(b-k)x-ka=0$ has exactly one negative root.

Accepted Answer

$a = -1$, $b = -3$; exactly one negative root when $k > 0$

FM 395 P1 Q1

FM 395 P1 Q1

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