Question

A curve with equation of the form $y=a{x}^4+b{x}^3+cx+d$ has zero gradient at the point $(0,1)$ and also touches the x-axis at the point $(-1,0)$ then

• A. $a=3$
• B. $b=4$
• C. $c+d=1$
• D. for $x<-1$ the curve has a negative gradient

Answer 1

The correct option is D for $x<-1$ the curve has a negative gradient

given $y=a{x}^4+b{x}^3+cx+d$

$y^{}{}^{\prime }=4a{x}^3+3b{x}^2+c=0$ at (0,1)

so $c=0$

curve passes through point (-1,0)

so $0=a-b-c+d=>a+d=b$

here $y^{}{}^{\prime }<0$ for $x<-1$