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

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

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

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

Correct Option: , 3

Solution:

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