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Math Help - Verify that the slope of the tangent at this point is zero.

  1. #1
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    Verify that the slope of the tangent at this point is zero.

    From: Mr.DJ7 - Public Text WindowHello, I'm having trouble on this question, because my book didn't describe the solution properly:

    For Each Function the point given is the maximum or minimum.
    Use the difference quotient to verify that the slope of the tangent at this point is zero.

    1. f(x)=0.5x^2 +6x + 7.5 Cords (-6, -10.5)

    I also need help with this question:

    Use an algebraic strategy to verify that the point given for each function is either a maximum or a minimum.

    f(x)= x^2 - 4x + 5 Cords (2, 1)
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  2. #2
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    for the first problem ...

    \lim_{x \to -6} \frac{f(x) - f(-6)}{x - (-6)}

    \lim_{x \to -6} \frac{\left(\frac{x^2}{2} + 6x + 7.5\right) + 10.5}{x +6}

    \lim_{x \to -6} \frac{\frac{x^2}{2} + 6x + 18}{x + 6}

    \frac{1}{2} \lim_{x \to -6} \frac{x^2 + 12x + 36}{x + 6}

    \frac{1}{2} \lim_{x \to -6} \frac{(x+6)^2}{x + 6}

    \frac{1}{2} \lim_{x \to -6} (x + 6) = 0

    for the second problem ...

     f(x)= x^2 - 4x + 5

    remember how to find the coordinates for the vertex of a parabola?
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  3. #3
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    Thankyou.
    But I'm still kind of confused,
    how did you get

    <br />
\frac{1}{2} \lim_{x \to -6} \frac{x^2 + 12x + 36}{x + 6}<br />

    And for the second one I Just substitute Y with f(x) and X with x?
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  4. #4
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    factor out 1/2 ...

    \frac{x^2}{2} + 6x + 18 = \frac{1}{2}(x^2 + 12x + 36)


    for the second problem, does x = \frac{-b}{2a} ring a bell?
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