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Let f(x)=12x2. (a) If we wish to find the point Q=(x0,y0) on the graph of f(x)...

Let f(x)=12x2. (a) If we wish to find the point Q=(x0,y0) on the graph of f(x) that is closest to the point P=(4,1), what is the objective function? (Hint: optimize the square of the distance from P to Q.)

Objective function: O(x)=O(x)=____________________________

(b) Find the point Q=(x0,y0)Q=(x0,y0) as described in part (a). Box your final answer.

(c) Verify that the line connecting PP to QQ is perpendicular to the line tangent to f(x)f(x) at QQ. Hint: recall that the lines y=m1x+b1y=m1x+b1 and y=m2x+b2y=m2x+b2 are perpendicular if m1=−1m2m1=−1m2 (assuming neither are horizontal or vertical).

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