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I have here quite a easy optimization problem, however I can't figure out how to solve it. Given a definite integral from a to b. I need to find values a and b such that the value of the integral is largest. How do I solve su

Schidu Luca
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  • If $\int_{a}^{b}f(x)dx$ has a closed form on $a$ and $b$, you should solve: $min\hspace{2mm} F(a)-F(b)$, whose solution depends on $F$. – Alex Silva Jan 04 '15 at 18:41
  • You need to give some more information. – copper.hat Jan 04 '15 at 18:54
  • It's a simple quadratic integral x^2 + x - 6 – Schidu Luca Jan 04 '15 at 18:55
  • In this case $$\int_{a}^{b}(x^2+x-6) dx = \left(\frac{1}{3}b^3 + \frac{1}{2}b^2 -6b \right) - \left(\frac{1}{3}a^3 + \frac{1}{2}a^2 -6a \right).$$ Thus, the integral is maximum if $a \rightarrow -\infty$ and $b \rightarrow \infty$. – Alex Silva Jan 04 '15 at 19:46

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Solve the integral with $a$ and $b$ as the limits, and then depending on the form of the answer there are different ways to maximise it with respect to $a$ and $b$. Either you can see the answer immediately, or you differentiate and find values of $a$ and $b$ for which the derivative is zero and find which of those is the maximum.

user141592
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