Q. 33 Find the point on the graph of t... [FREE SOLUTION]

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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.
$f (x) = x^{2} 鈭� 2 x + 1$ and the point $(1, 2)$

Short Answer

Expert verified

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

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,

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

Now,

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

Step 3. Again,

$\begin{aligned} D^{鈥�} (x) = 0 \\ \frac{2 x^{3} 鈭� 6 x^{2} + 3 x + 1}{\sqrt{x^{4} 鈭� 4 x^{3} + 3 x^{2} + 2 x + 2}} = 0 \\ 2 x^{3} 鈭� 6 x^{2} + 3 x + 1 = 0 \\ x = 1, \frac{2 卤 \sqrt{6}}{2} \end{aligned}$

Putting $\frac{2 卤 \sqrt{6}}{2} in f (x) = x^{2} 鈭� 2 x + 1$

\begin{array}{ll} f (x) = 1.75 鈥呪赌呪赌呪赌� [x = \frac{2 + \sqrt{6}}{2}] \\ f (x) = 1.75 鈥呪赌呪赌呪赌� [x = \frac{2 鈭� \sqrt{6}}{2}] \end{array}

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

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91影视

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.
$f (x) = x^{2} 鈭� 2 x + 1$ 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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91影视

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鈭�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

Statistics

Pure Maths

Applied Mathematics

Calculus

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

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.
$f (x) = x^{2} 鈭� 2 x + 1$ and the point $(1, 2)$