Question

Consider the polynomial $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=\mid s$.
The area bounded by the curve $y=f(x)$ 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. $\left(0,\frac{21}{64}\right)$

Answer 1

Answer: (A)

$-\frac{3}{4}<s<-\frac{1}{2}$
$\frac{1}{2}<t<\frac{3}{4}$
$\underset{0}{\overset{1/2}{\int }}\left(4x^3+3x^2+2x+1\right)dx<$ area $<\underset{0}{\overset{3/4}{\int }}\left(4x^3+3x^2+2x+1\right)dx$
${\left[x^4+x^3+x^2+x\right]}_0^{1/2}<$ area $<{\left[x^4+x^3+x^2+x\right]}_0^{3/4}$
$\frac{1}{16}+\frac{1}{8}+\frac{1}{4}+\frac{1}{2}<$ area $<\frac{81}{256}+\frac{27}{64}+\frac{9}{16}+\frac{3}{4}$
$\frac{15}{16}<$ area $<\frac{525}{256}$