Question

The curve $y=a{x}^3+b{x}^2+cx+5$ touches the x-axis at $P(-2,0)$ and cuts the y-axis at a point $Q$ where its gradient is $3$. Then the value of $a,b,c$ is

• A. $A=\frac{9}{2}$, $B=\frac{1}{2}$, $C=\frac{3}{5}$
• B. $A=\frac{-1}{2}$, $B=\frac{-3}{4}$, $C=3$
• C. $A=\frac{9}{2}$, $B=\frac{-3}{4}$, $C=3$
• D. $A=\frac{-9}{2}$, $B=\frac{1}{2}$, $C=\frac{-3}{5}$

Answer 1

The correct option is B $A=\frac{-1}{2}$, $B=\frac{-3}{4}$, $C=3$

$y=a{x}^3+b{x}^2+cx+5$ $P(-2,0)$ at $y$ axis

differentiate $y$ wrt $x$ $y=5@\varphi (0,5)m=3$

${\left(\frac{dy}{dx}\right)}_{@\varphi }=3a{x}^2+2bx+c$

${\left(\frac{dy}{dx}\right)}_{@(0,5)}=3a(0{)}^2+2b(0)+c=3$

$c=3$

${\left(\frac{dy}{dx}\right)}_{@P=(-2,0)}=3a(-2{)}^2+2b(-2)+3=0$

$12a-4b+3=0....\left(1\right)$

$-8a+4b-1=0.....\left(2\right)$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$a=-1/2b=-3/4$