Chapter 3: Q. 33 (page 288)

Find the point on the graph of the function $f$ that is closest to the point $(a, b)$ by minimizing the square of the distance from the graph to the point.
$f (x) = x^{2} - 2 x + 1$ and the point $(1, 2)$

Short Answer

Expert verified

Ans: The point is $(\frac{2 + \sqrt{6}}{2}, 1.75); (\frac{2 - \sqrt{6}}{2}, 1.75)$

Step by step solution

Step 1. Given information.

given expression,

$f (x) = x^{2} - 2 x + 1$

Step 2. The objective is to find a point closest to the function that is closest to the point by minimizing the graph from the distance to the point.

Let, the point be $(x, y)$ .

The distance between $(1, 2)$ ) and $(x, y)$ is,

$\begin{aligned} D (x) = \sqrt{(2 - y)^{2} + (1 - x)^{2}} \\ D (x) = \sqrt{{(2 - x^{2} + 2 x - 1)}^{2} + (1 - x)^{2}} \\ D (x) = \sqrt{{(- x^{2} + 2 x + 1)}^{2} + 1 - 2 x + x^{2}} \\ D (x) = \sqrt{x^{4} + 4 x^{2} + 1 - 4 x^{3} + 4 x - 2 x^{2} + 1 - 2 x + x^{2}} \\ D (x) = \sqrt{x^{4} - 4 x^{3} + 3 x^{2} + 2 x + 2} \end{aligned}$

Now,

$\begin{aligned} D (x) = \sqrt{x^{4} - 4 x^{3} + 3 x^{2} + 2 x + 2} \\ D^{'} (x) = \frac{2 x^{3} - 6 x^{2} + 3 x + 1}{\sqrt{x^{4} - 4 x^{3} + 3 x^{2} + 2 x + 2}} \end{aligned}$

Step 3. Again,

$\begin{aligned} D^{'} (x) = 0 \\ \frac{2 x^{3} - 6 x^{2} + 3 x + 1}{\sqrt{x^{4} - 4 x^{3} + 3 x^{2} + 2 x + 2}} = 0 \\ 2 x^{3} - 6 x^{2} + 3 x + 1 = 0 \\ x = 1, \frac{2 \pm \sqrt{6}}{2} \end{aligned}$

Putting $\frac{2 \pm \sqrt{6}}{2} in f (x) = x^{2} - 2 x + 1$

\begin{array}{ll} f (x) = 1.75 [x = \frac{2 + \sqrt{6}}{2}] \\ f (x) = 1.75 [x = \frac{2 - \sqrt{6}}{2}] \end{array}

Therefore, the point is $(\frac{2 + \sqrt{6}}{2}, 1.75); (\frac{2 - \sqrt{6}}{2}, 1.75)$

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Find the point on the graph of the function $f$ that is closest to the point $(a, b)$ by minimizing the square of the distance from the graph to the point.
$f (x) = x^{2} - 2 x + 1$ and the point $(1, 2)$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to find a point closest to the function that is closest to the point by minimizing the graph from the distance to the point.

Step 3. Again,

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Find the point on the graph of the functionf that is closest to the point (a,b) by minimizing the square of the distance from the graph to the point. f(x)=x2−2x+1and the point(1,2)

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to find a point closest to the function that is closest to the point by minimizing the graph from the distance to the point.

Step 3. Again,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Find the point on the graph of the function $f$ that is closest to the point $(a, b)$ by minimizing the square of the distance from the graph to the point.
$f (x) = x^{2} - 2 x + 1$ and the point $(1, 2)$