anonymous
  • anonymous
Suppose that the function f has a continuous second derivative for all x, and that f(0)=2, f'(0)=3, and f"(0) =0. Let g be a function whose derivative is given by g'(x)=e^-2x(3f(x)+2f'(x)) for all x. Show that g"(x)=e^-2x(-6f(x)-f'(x)+2f"(x)) Does g have a local maximum at x=0?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
is that e to the (-2x )then back down, times the rest of that, or is it all raised from e?
anonymous
  • anonymous
back down then the rest of it.
anonymous
  • anonymous
let me look at some more, I'll come back in a few minutes

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anonymous
  • anonymous
okay thankyou!
anonymous
  • anonymous
just to make sure, we are trying to prove that the second derivative of g(x) has the local max at x= zero?
anonymous
  • anonymous
we're trying to see if g(x) has a local maximum at x=0 i think,..
anonymous
  • anonymous
okay, that make a whole lot more sense. brb
anonymous
  • anonymous
okay!
anonymous
  • anonymous
so, take that second derivative equation and sub in 0 for x, and all of the f(0) values. this should give you g''(0)= -15, because e^0 = 1. Whne you integrate g''(0)=15, g'(x) = 15x, @ 0 = 0, therefore you have a critical point. so sub in 1 and -1 on a number line for your first derivative and you can see that the slopes change from + to - at 0, therefore there is a local max.
anonymous
  • anonymous
okay ill try and do it & let you know if i get stuck. :)
anonymous
  • anonymous
I don't know why they gave you that other g'(x), but if g''(x)= wht you worte, then go back from there.
anonymous
  • anonymous
wrote
anonymous
  • anonymous
so do the antiderivative of the first derivative to get my g(x)?
anonymous
  • anonymous
right
anonymous
  • anonymous
okay. ima finish working it.
anonymous
  • anonymous
now it looks like the second deriv and the first derivs you're given are different, but if you find the first antideriv of the second, then the antideriv of that, it would give you g(x) from g''(x). It looks like you just have to prove that g(x) has a max, so you only need g'(x) to do it.
anonymous
  • anonymous
okay and how do i do that using only the g'?
anonymous
  • anonymous
only the g'(x) that they gave you doesn't work out. It only works using the g''(x) from what I could see. The g'(x) is = 12 at 0, and therefore could not be a critical value. It looks like you have to get g'(x) from g''(x), unless the g'(x) you wrote is different from what you have in the book or what not.

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