Question

definite integral help

Answer 1

The integrand will be the derivative of \(F\)

Answer 2

So just find it's roots

Answer 3

\[\cos(9x ^{5/2})\] ?

Answer 4

Roots of \(\cos\) are at \(\pi/2\) and \(3\pi/2\).

Answer 5

So you can set: \[
9x^{5/2} = \pi/2
\]

Answer 6

so
\[x ^{5/2}=\frac{ \pi }{ 18 }\]

Answer 7

Yeah, you'll be solving for \(x\).

Answer 8

Since \(9x^{5/2}\) is an always increasing function, we know that it will hit \(\pi/2\) before it hits \(3\pi/2\), if you were curious.

Answer 9

Next step would be to take the second derivative of \(F\) to figure out if we have a maximum or minimum.

Answer 10

what does x =?

Answer 11

Do you know how to get rid of the exponent \(5/2\)?

Answer 12

no

Answer 13

How would you get rid of the exponent \(3\)?

Answer 14

take the 3rd root

Answer 15

so i would take the 5th root and square both sides

Answer 16

Yeah, but the 3rd root is also an exponent

Answer 17

\[
\sqrt[n]{x} = x^{1/n}
\]

Answer 18

So we can use this property to get rid of \(5/2\) by finding the reciprocal.

Answer 19

2/5

Answer 20

so\[x=(\frac{ \pi }{ 2 })^{2/5}\]

Answer 21

Where did the \(9\) go?

Answer 22

i mean \[x=(\frac{ \pi }{ 18 })^{2/5}\]

Answer 23

Ok

Answer 24

so \[\frac{ \pi^\frac{ 2 }{ 5 } }{ 18^{2/5} }\]

Answer 25

You've really gotten into this... ok

Now can you get the second derivative?