If $f(x)$ is a quadratic in $x$, then $\int_0^1 f(x) d x$ is equal to

Question

If $f(x)$ is a quadratic in $x$, then $\int_0^1 f(x) d x$ is equal to, $\frac{1}{6}\left[\mathrm{f}(0)+4 \mathrm{f}\left(\frac{1}{2}\right)+\mathrm{f}(1)\right]$, $\frac{1}{6}\left[4 \mathrm{f}(0)+\mathrm{f}\left(\frac{1}{2}\right)+\mathrm{f}(1)\right]$, $\frac{1}{6}\left[\mathrm{f}(0)+\mathrm{f}\left(\frac{1}{2}\right)+4(1)\right]$, $\frac{1}{6}\left[f(0)+f\left(\frac{1}{2}\right)+f(1)\right]$

MARKS App · Accepted Answer

If $f(x)=a x^2+b x+c$, $\begin{aligned} & \int_0^1 f(x) d x=\frac{1}{6}(2 a+3 b+6 c) \ & f(0)=c, f\left(\frac{1}{2}\right)=\frac{a}{4}+\frac{b}{2}+c, f(1)=a+b+c \ & \Rightarrow f(0)+4 f\left(\frac{1}{2}\right)+f(1)=2 a+3 b+6 c \end{aligned}$

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