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find the maximim value of the integral (3/4 -x-X^2)dx lim b to a b is greater than a

6 years ago

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Ans:
$I = \int_{b}^{a}(\frac{3}{4}-x-x^{2})dx$
$I = (\frac{3x}{4}-\frac{x^{2}}{2}-\frac{x^{3}}{3})_{b}^{a}$
$I = \frac{3(a-b)}{4}-\frac{a^{2}-b^{2}}{2}-\frac{a^{3}-b^{3}}{3}$
$I = \frac{b(4b^{2}+6b-3)-a(4a^{2}+6a-3)}{12}$
$b>a$
$b = a + h, h\geq 0$
$I = \frac{(a+h)(4(a+h)^{2}+6(a+h)-3)-a(4a^{2}+6a-3)}{12}$
$I = \frac{4h^{3} + (12a+6)h^{2} + (12a^{2}+12a-3)h }{12}$
$\frac{\partial I}{\partial h} = \frac{12h^{2} + 2(12a+6)h + (12a^{2}+12a-3) }{12}= 0$
Find ‘h’ from here:
$12h^{2} + 2(12a+6)h + (12a^{2}+12a-3) = 0$
$h = \frac{-2(12a+6)\pm \sqrt{288}}{24}$
$h = -a + \frac{\sqrt{2}-1}{2}$
$b = h +a = \frac{\sqrt{2}-1}{2}$
Thanks & Regards
Jitender Singh
IIT Delhi
2 years ago

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