If $\int_a^b e^{cosx}(380-x-x^2)dx$ attains its maximum value for some $(a,b)$ such that $a\leq b$, then the value of $b-a$ is?
My attempt
Factorized the integrand to reduce it to $\int_a^b e^{cosx}(20+x)(19-x)dx$, now my first instinct was to take $b-a=c$, treat $a$ as a constant and then use the Newton-Leibniz formula to differentiate the integral $\int_a^{a+c} e^{cosx}(20+x)(19-x)dx$ with respect to c, which gives me values of $a+c$ but doesn't seem to lead anywhere. Along with this taking $a$ as a constant is pretty random and has no basis. I suspect some property of definite integration may be involved but am unable to find something to lead me anywhere.
Hints or solutions are appreciated
$\endgroup$ 21 Answer
$\begingroup$The question is about the global maximum of the two variable function$$ F(a,b)=\int_a^b e^{\cos x}(380-x-x^2)dx. $$
Since this is a differentiable function in an open set ($\mathbb{R}^2$), the maximum must occur in a stationary point, which satisfies the conditions
$$F'_a = - e^{\cos a}(380-a-a^2) = 0, \quad F'_b=e^{\cos b}(380-b-b^2) = 0.$$
We can check that the possible values for $a,b$ are $-20,19$. We can rule out the case $a=b$ because $F(a,a)=0$ is not the maximum e conclude that the maximum is attained for $a=-20, b=19$. So, $b-a = 39$, and the maximum is approximately $12457.1$.
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