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Math Help - Estimating value of function using first and second derivative

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    Estimating value of function using first and second derivative

    The function f is twice differentiable, and has no points of inflection. If f(6)=3, f'(6)=-1/2, and f''(6)=-2 which of the following could be the value of f(7)? 2, 2.5, 2.9, 3, 4.
    I can rule out the last two options easily but I feel like I'm missing an important detail than besides the function is ccu and decreasing. The answer's two, but I don't see why it could not be 2.5.
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    Quote Originally Posted by eulcer View Post
    The function f is twice differentiable, and has no points of inflection. If f(6)=3, f'(6)=-1/2, and f''(6)=-2 which of the following could be the value of f(7)? 2, 2.5, 2.9, 3, 4.
    I can rule out the last two options easily but I feel like I'm missing an important detail than besides the function is ccu and decreasing. The answer's two, but I don't see why it could not be 2.5.
    Here is one way to look at it

    You know the function goes though the point (6,3) the slope is -1/2 and is is getting smaller (more negative).

    If that is the cases the biggest the function could be is 3-1/2=2.5, but since the slope is getting more negative it must be less than that.
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