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Chapter 9: Problem 29

Find the points on the graph of the function that are closest to the given point. \(f(x)=x^{2}, \quad\left(2, \frac{1}{2}\right)\)

Short Answer

Expert verified
The points on the graph of the function that are closest to the point \((2, \frac{1}{2})\) are given by \((x_1, f(x_1))\) and \((x_2, f(x_2))\), where \(x_1\) and \(x_2\) are the solutions of the quadratic equation from step 4.

Step by step solution

01

Formulate the Distance Function

Since we're going to find the points on the graph that are closest to a specific point, the first thing we should do is to formulate a distance function. In Euclidean geometry, the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\). Now, replace \(y_1\) with the function \(f(x)\), and \(x_2\), \(y_2\) with the point coordinates. This leads us to the function \(D(x) = \sqrt{(x - 2)^2 + (x^2 - \frac{1}{2})^2}\).
02

Simplify the Function

Finding derivative of a square root function could be quite complex. In order to avoid this, we recognize that minimizing the square of the distance is equivalent to minimizing the distance itself. Hence, redefine the function as \(D_s(x) = (x - 2)^2 + (x^2 - \frac{1}{2})^2\). This won't affect the point where minimum occurs.
03

Derive the Function

Differentiate \(D_s(x)\): \(D_s'(x) = 2(x - 2) + 2(x)(2x - \frac{1}{2})\).
04

Set the Derivative to Zero

To find the points where minimum occurs, set the \(D_s'(x)\) to zero and solve for \(x\). Thus, \(2(x - 2) + 2(x)(2x - \frac{1}{2}) = 0\). This is a quadratic equation, which can be solved by factoring or using the quadratic formula.
05

Solve for x

The solutions to the quadratic equation from the previous step are \(x_1\) and \(x_2\). Plug these values into \(f(x)\) to get the y-coordinates. Thus, the closest points are \((x_1, f(x_1))\) and \((x_2, f(x_2))\) on the graph.

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Most popular questions from this chapter

The state game commission introduces 30 elk into a new state park. The population \(N\) of the herd is modeled by \(N=[10(3+4 t)] /(1+0.1 t)\) where \(t\) is the time in years. (a) Find the size of the herd after 5,10 , and 25 years. (b) According to this model, what is the limiting size of the herd as time progresses?

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