Question

Calculus and Geometry How close does the curve \(y=\sqrt{x}\) come to the point \((3 / 2,0) ?[\)Hint: If you minimize the square of the distance, you can avoid square roots.

Short answer

The curve \(y=\sqrt{x}\) comes as close as \( \sqrt{5 / 4} \) or approximately 1.118 to the point \((3 / 2,0)\).

Step-by-step solution

Assign the Function

Assign the function \(y=\sqrt{x}\). Next, the distance squared between a point on the curve and the given point \((3 / 2,0)\) would be \((x - 3 / 2)^2+ (\sqrt{x} - 0)^2\), which simplifies to \((x - 3 / 2)^2 + x\). This is also our function. Let’s denote it as \(F(x)\), thus, \(F(x)=(x - 3 / 2)^2 + x\).

Find the Derivative

Find the derivative of \(F(x)\). The derivative \(F'(x)\) would be \(2(x - 3 / 2) + 1\).

Set the Derivative Equal to Zero

Set the derivative equal to zero and solve for \(x\). Solving \(2(x - 3 / 2) + 1 = 0\) gives the value \(x = 1\).

Check for Minimum

To ensure that \(x = 1\) gives the minimum value of the function, we can use the second derivative test. Calculate the second derivative \(F''(x)\) which gives \(2\). Since this is positive, we know that \(x = 1\) gives a minimum.

Calculate the Minimum Distance

Finally, plug \(x = 1\) into the original distance squared formula to find \(F(1)\), which is \((1 - 3 / 2)^2 + 1 = 1 / 4 + 1 = 5 / 4\).