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Find the shortest distance from the point (0, 17/4) to the parabola y…
Question
Find the shortest distance from the point (0, 17/4) to the parabola y = x^2.
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Step-by-step solution
A point on the parabola is (x, x^2).
The squared distance to (0, 17/4) is D = x^2 + (x^2 - 17/4)^2.
Let u = x^2 >= 0, so D(u) = u + (u - 17/4)^2.
Then D'(u) = 1 + 2(u - 17/4) = 0 gives u = 17/4 - 1/2 = 15/4 (which is >= 0, valid).
Then D = 15/4 + (15/4 - 17/4)^2 = 15/4 + (1/2)^2 = 15/4 + 1/4 = 4, so the distance is sqrt(4) = 2.
Final answer2
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