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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