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A function f[x] and a point x=a. If f'[a]>0, is it possible f[a]<=f[x]

  1. Sep 25, 2005 #1

    CAG

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    You've got a function f[x] and a point x=a.
    If f'[a]>0, is it possible that f[a]<=f[x] a for all other x's?
    Why?

    Having a problems visualize this, any help would be great
     
  2. jcsd
  3. Sep 25, 2005 #2
    If f'[a] is positive, the function is increasing at a. For f[a]<=f[x] to be true, "a" needs to be a maxima.
     
  4. Sep 25, 2005 #3
    Do you mean minima?

    ...and by the way, if it was a minima, f'(a) would be 0. So since f'(a)>0, what does that tell you about x<a?
     
  5. Sep 25, 2005 #4
    I good example would be [tex] f(x) = \sin x [/tex] the derivative of which is [tex] f'(x) = \cos x [/tex]. If [tex] a = 0 [/tex] then [tex] f'(0) = \cos 0 = 1 [/tex] indicating the function is rising. However, since sine is a repeating function, it will go to a maximum of 1, and fall down again. So f(a) <= f(x) for all x > a is not true if f(x) repeats on the segment a < x < inf.
     
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