Maximum of integral?

Calculus Level 4

f ( x ) = 1 2 x sin ( a ) e 2 a + 1 d a \large f(x)=\int _{-\frac 12}^x \sin (a) e^{2a+1}da

For function f ( x ) f(x) as defined above, find the value of x [ 0 , 2 π ] x \in [0, 2\pi] for which f ( x ) f(x) is a maximum. Give your answer to 3 significant figures.


The answer is 3.14.

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1 solution

Chew-Seong Cheong
Sep 17, 2018

f ( x ) f(x) is an extremum when f ( x ) = e 2 x + 1 sin x = 0 f'(x) = e^{2x+1}\sin x = 0 or a = 0 , π a = 0, \pi for x [ 0 , 2 π ) x \in [0, 2\pi) . Since f ( x ) = ( 2 x + 1 ) e 2 x + 1 sin x + e 2 x + 1 cos x f''(x) = (2x+1)e^{2x+1}\sin x + e^{2x+1}\cos x . Then f ( 0 ) = e > 0 f''(0) = e > 0 f ( 0 ) \implies f(0) is a minimum; and f ( π ) = e 2 π + 1 < 0 f''(\pi) = - e^{2\pi + 1} < 0 f ( x ) \implies f(x) is a maximum when x = π 3.14 x = \pi \approx \boxed{3.14} .

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