If for all real triplets

Question:

If for all real triplets $(a, b, c), f(x)=a+b x+c x^{2} ;$ then $\int_{0}^{1} f(x) d x$ is equal to:

  1. (1) $2\left\{3 f(1)+2 f\left(\frac{1}{2}\right)\right\}$

  2. (2) $\frac{1}{2}\left\{f(1)+3 f\left(\frac{1}{2}\right)\right\}$

  3. (3) $\frac{1}{3}\left\{f(0)+f\left(\frac{1}{2}\right)\right\}$

  4. (4) $\frac{1}{6}\left\{f(0)+f(1)+4 f\left(\frac{1}{2}\right)\right\}$


Correct Option: , 4

Solution:

$\int_{0}^{1}\left(a+b x+c x^{2}\right) d x=a x+\frac{b x^{2}}{2}+\left.\frac{c x^{3}}{3}\right|_{0} ^{1}=a+\frac{b}{2}+\frac{c}{3}$

Now, $f(1)=a+b+c, f(0)=a$ and $f\left(\frac{1}{2}\right)=a+\frac{b}{2}+\frac{c}{4}$

Now, $\frac{1}{6}\left(f(1)+f(0)+4 f\left(\frac{1}{2}\right)\right)$

$=\frac{1}{6}\left(a+b+c+a+4\left(a+\frac{b}{2}+\frac{c}{4}\right)\right)$

$=\frac{1}{6}(6 a+3 b+2 c)=a+\frac{b}{2}+\frac{c}{3}$

Hence, $\int_{0}^{1} f(x)=\frac{1}{6}\left\{f(0)+f(1)+4 f\left(\frac{1}{2}\right)\right\}$

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