Paragraph: Consider the polynomial f(x)=1+2 x+3 x^2+4 x^3 . Let s...

Paragraph: 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=|s|

.Question: The area bounded by the curve

y=f(x)

and the lines

x=0, y=0

and

x=t

, lies in the interval Paragraph: 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

Options

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

Correct answer

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

Step-by-step solution

\int_0^{1 / 2} f(x) d x < \int_0^t f(x) d x < \int_0^{3 / 4} f(x) d x

Now,

\int f(x) d x

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\begin{gathered} =\int\left(1+2 x+3 x²+4 x³\right) d x \\ =x+x²+x³+x⁴ \\ \Rightarrow \quad \int_0^{1 / 2} f(x) d x=\frac{15}{16}>\frac{3}{4}\\ \int_0^{3 / 4} f(x) d x=\frac{530}{256}< 3 \end{gathered}

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