Question

If $(1,3)$ is the point of inflection of the curve $y=a{x}^3+b{x}^2,$ then the value of $4(a^2+b^2)$ is

• A. 90.0
• B. 90
• C. 90.00
• D. 90.000

Answer 1

Answer: (A), (B), (C), (D)

Given, $y=a{x}^3+b{x}^2$
And P.O.I. $\equiv (1,3)$
$(1,3)$ lies on the curve
$\Rightarrow a+b=3\cdots \left(i\right)$
$y^{\prime }=3a{x}^2+2bx$
$y^{\prime \prime }=6ax+2b$
Also we have $y^{\prime \prime }\left(1\right)=0$
$\Rightarrow 3a+b=0\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$,
$a=-\frac{3}{2},b=\frac{9}{2}$
Now, $4(a^2+b^2)=4(\frac{9}{4}+\frac{81}{4})=90$