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Math Help - Help Asap Please

  1. #1
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    Help Asap Please

    P is a point on the parabola f(x) = 1-x^2. A tangent is drawn at point P cuts x axes and y axes to form a triangle. Find the coordinates of P that will minimize the area of the triangle
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  2. #2
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    Hello, pita!

    P is a point on the parabola: f(x) \:= \:1-x^2
    A tangent is drawn at point P cuts x-axis and y-axis to form a triangle.
    Find the coordinates of P that will minimize the area of the triangle.

    The point P has coordinates: . \left(p,\:1-p^2\right)

    The derivative is: . f'(x) \:=\:-2x
    The tangent at P has slope: . m \:=\:-2p

    The tangent at P has the equation: . y - (1-p^2) \;=\;-2p(x-p)
    . . which simplifies to: . y \;=\;-2px + p^2+1

    \text{It has }x\text{-intercept }\left(\frac{p^2+1}{2p},\:0\right)\;\text{ and }\;y\text{-intercept }\left(0,\:p^2+1\right)


    The area of the triangle is: . A \;=\;\frac{1}{2}\left(\frac{p^2+1}{2p}\right)\left  (p^2+1\right) \;=\;\frac{1}{4}\left(\frac{p^4 + 2p^2 + 1}{p}\right)

    Therefore: . A \;=\;\frac{1}{4}\left(p^3 + 2p + p^{-1}\right) .is the function to be maximized.

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