Olivia
Lv 4
Olivia asked in Science & MathematicsMathematics · 1 decade ago

True or False: If f ''(x) has an absolute minimum at x = a, then f '(a) must exist and have the value 0..

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  • Anonymous
    1 decade ago
    Favorite Answer

    True, if f' is continuous and differentiable at a. But it need not be.

  • cj k
    Lv 4
    1 decade ago

    False for the value of 0, I think that f'(a) does have to exist though

    for example:

    f''(x)=x^2+1

    min @x=0

    f'(x)=(1/3)x^3+x+b

    f'(0)=0+0+b not 0 except if b=0

  • 1 decade ago

    Are you sure you are asking that correctly? I think the question should be be in regards to f(a) (the function) and not f''(x) (the second derivative).

    If the question is about f(x) and f(x) is at a minimum at f(a) (absolute or local for that matter), then the slope of f(a) -- i.e. a line drawn tangent to the curve at a will be flat, correct?

    Well, the first derivative of a function IS the slope of that tangent line at a given point, so therefore the f'(a) would be zero because that line at any minima or maxima is flat.

  • 1 decade ago

    no

    it just means that the value of f' at a is the steepest possible slope of the graph

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  • 1 decade ago

    You don't really expect any honest answer for this do you lol

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