Maximum value of a given integral
For $a,b∈R$, and $b>a$, what is the maximum possible value of the integral $$\int_{a}^{b}7x-x^2-10dx?$$ I have no idea how to solve it. Please help me to solve this. Thanks in advance.
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$\begingroup$If you integrate over the region where $7x-x^2-10$ is negative, the value will be less.
If you integrate over the region where the $7x-x^2-10$ is positive, the value will increase.
Hence we want to solve the problem of $$7x-x^2-10\geq 0$$
$$x^2-7x+10 \leq 0$$
$$(x-2)(x-5) \leq 0$$
One should be able to tell the value of $a$ and $b$ from on top and substitue it to evaluate the maximum value.
$\endgroup$ 1 $\begingroup$Let $f(x)=7x-x^2-10$ we have $f(x)=-(x-5)(x-2)$ thus $2,5$ are the roots of this function.Between these roots the curve is above x-axis.Thus $a=2,b=5$ thus maximum possible are is $\frac{9}{2}$.
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