JEE Advanced Calculus problem

Calculus Level 5

Determine \(f(x) = ax^2+bx+c\), where \(a > 0\), whose roots are \(\phi_1\) and \(\phi_2\) where \(0 < \phi_2 < \phi_1\)

If, for some real value \(\alpha < \beta\) where \(\alpha+\beta = 5\), \(\displaystyle\int_{0}^{\alpha}f(x)dx = 0\) and \[\int_{0}^{\phi_1}f(x)dx+\int_{\phi_2}^{\beta}f(x)dx=|\int_{\phi_1}^{\phi_2}f(x)dx|\] The value of \(x\) such that \(f(x)\) is minimum can be written in the form of \(\displaystyle\frac{p}{q}\), where \(p\) and \(q\) are coprime. Find \(p+q\)

Original Question

-Also note that alpha is less than beta.

  • The answer will be in form of \( \frac{p}{q} \).

  • Find p+q.


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