Question

Consider the polynomial $f\left(x\right)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of $f\left(x\right)$ and let $t=|s|$.

The area bounded by the curve $y=f\left(x\right)$ and the lines $x=0,y=0$ and $x=t$, lies in the interval

• A. $\left(\frac{3}{4},3\right)$
• B. $\left(\frac{21}{64},\frac{11}{16}\right)$
• C. $(9,10)$
• D. $(0,\frac{21}{64})$

Answer 1

Answer: (A)

$\left(\frac{3}{4},3\right)$

$-\frac{3}{4}<s<-\frac{1}{2}$
$\frac{1}{2}<t<\frac{3}{4}$
${\int }_0^{1/2}(4x^3+3x^2+2x+1)dx<$ area $<\displaystyle

${\int }_0^{3/4}(4x^3+3x^2+2x+1)dx$

$[x^4+x^3+x^2+x{]}_0^{1/2}<area<[x^4+x^3+x^2+x{]}_0^{3/4}$
$\frac{1}{16}+\frac{1}{8}+\frac{1}{4}+\frac{1}{2}<area<\frac{81}{256}+\frac{27}{64}+\frac{9}{16}+\frac{3}{4}$

$\frac{15}{16}<area<\frac{525}{256}$