Please show step by step work. If you choose to do it on paper and upload a pict

Question

Please show step by step work.

If you choose to do it on paper and upload a picture, please make sure that it is legible.

A wire is bent to fit the curve y = 1 - x2. A string is stretched from the origin to a point (x, y) on the curve. Use the two methods indicated to find the coordinates of the point that will minimize the length of the string. We want to minimize the distance d = y root x2 + y2 from the origin to the point (x, y); this is equivalent to minimizing f = d2 = x2 + y2. However, x and y are not independent; they arc related by the equation of the curve. That extra relation between the variables is what is often referred to as a constraint. Problems involving constraints occur frequently in real-life applications. (6 pts.) Method 1: Elimination. Start by algebraically eliminating y. (I.e., solve fory and substitute).

Explanation / Answer

f=d^2=x^2+y^2=x^2+(1-x^2)^2=x^4-x^2+1

now for maximixation

df/dx=4x^3-2x=0

==> either x=0 or x=+1/sqrt(2) or x=-1/sqrt(2)

d^2f/dx^2=(12x^2-2)

for a point to be a maxima d^2f/dx^2>0

so for the points

x=+1/sqrt(2)

d^2f/dx^2=4>0

and

x=-1/sqrt(2)

d^2f/dx^2=4>0

==> maximum value of distance is obtained at the points x=+1/sqrt(2) and x=-1/sqrt(2)

==> coordinates = (+1/sqrt(2), 1/2) , (-1/sqrt(2), 1/2)

==> coordinates = (0.707, 0.5) , (-0.707, 0.5)

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