Please show step by step work. It may seem like a lot, but when you read it, it\

Question

Please show step by step work. It may seem like a lot, but when you read it, it's actually not.

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. (8 pts.) Method 2; Implicit Differentiation. Suppose that it had not been possible to solve for y and substitute. We could still do the problem by using the fact that f = x2 + y2. df = or df / dx = From an equation like y = 1 - x2 relating x and y. we could find Jy in terms of dx even if the equation was not solvable fory. Here we get dy = Eliminating dy from df, we have df = or df = Now, all that is left is to minimize f, and then to simultaneously solve the resultant equation with the equation of the curve.

Explanation / Answer

df= 2x*dx + 2y*dy

or

df/dx=2x+2y*dy/dx

now from y=1-x^2

dy=-2x*dx

eliminating dy from df

==> df/dx= 2x+2y*(-2x)=(2x-4xy)=2x(1-2y)=2x(1-2+2x^2)

==> df/dx =2x(2x^2-1)

for maximizing f

df/dx=0

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

==> 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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