Question

Consider the polynomial $f ( x )=1+2 x +3 x ^{2}+4 x ^{3}$. 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)

$-\frac{3}{4} < s < -\frac{1}{2}$
$\frac{1}{2} < t < \frac{3}{4}$
$\int\limits_{0}^{1 / 2}\left(4 x^{3}+3 x^{2}+2 x+1\right) d x<$ area $<\int\limits_{0}^{3 / 4}\left(4 x^{3}+3 x^{2}+2 x+1\right) d x$
$\left[x^{4}+x^{3}+x^{2}+x\right]_{0}^{1 / 2}<$ 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 $<\frac{525}{256}$