Solve the problem.
-How close does the curve $y = \sqrt { x }$ come to the point $( 3,0 )$ ? (Hint: If you minimize the square of the distance, you can avoid square roots.)
A) The distance is minimized when $x = 2$; the minimum distance is 3 units.
B) The distance is minimized when $x = - \frac { 1 } { 3 }$; the minimum distance is $\sqrt { \frac { 11 } { 4 } }$ units.
C) The distance is minimized when $x = \frac { 5 } { 2 }$; the minimum distance is $\sqrt { \frac { 11 } { 4 } }$ units.
D) The distance is minimized when $x = 5$; the minimum distance is 3 units.

Question

Quizplus · Accepted Answer

The square of the distance between the point (3,0) and a point on the curve y = √x is given by (x-3)² + x. To find the minimum distance, we can minimize this expression. Taking the derivative of this expression with respect to x gives 2x-6+1 = 2x-5. Setting this derivative equal to 0 and solving for x gives x=5/2. To verify that this gives a minimum, we can take the second derivative of the expression and evaluate it at x=5/2, which gives 2. Since the second derivative is positive, we know that the expression has a minimum at x=5/2. Plugging this value of x back into the original expression for the square of the distance gives a minimum distance of √(11/4), which corresponds to choice C.

Solve the Problem y=xy = \sqrt { x }y=x​ Come to the Point

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