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Chapter 5: Problem 5

Find the point on the line \(y=2 x\) that is closest to the point (5,2)

Short Answer

Expert verified
The point on the line \(y = 2x\) that is closest to the point (5, 2) is \((\frac{9}{5}, \frac{18}{5})\).

Step by step solution

01

Distance formula

Let's denote the point on the line by (x, y). Since this point is on the line, it should satisfy the equation \(y = 2x\). The distance (d) between this point and the given point (5, 2) can be formulated using the distance formula: \[d = \sqrt{(x - 5)^2 + (y - 2)^2}\] Since y = 2x, the formula becomes \[d = \sqrt{(x - 5)^2 + (2x - 2)^2}\]
02

Simplify the distance function

To find the point where the distance is minimum, we need to minimize the distance function. Before doing that, let's simplify the distance function: \[d = \sqrt{(x - 5)^2 + (2x - 2)^2} = \sqrt{x^2 - 10x + 25 + 4x^2 - 8x + 4}\] Combining the terms, we get: \[d = \sqrt{5x^2 - 18x + 29}\]
03

Derive the distance function with respect to x

Since solving for the minimum distance directly is challenging, let's find the minimum of the square of the distance function instead (as this will also give us the minimum distance). Let \(d^2 = 5x^2 - 18x + 29\). Differentiating \(d^2\) with respect to x gives us: \[\frac{d(d^2)}{dx} = 10x - 18\]
04

Find the critical points

To find the critical points, we have to set the derived function equal to zero and solve for x: \[10x - 18 = 0\] Solving for x, we find that: \[x = \frac{18}{10} = \frac{9}{5}\]
05

Find the corresponding y-coordinate

Now that we have found the x-coordinate, we can find the corresponding y-coordinate by plugging the value into the line equation: \[y = 2x = 2 \cdot \frac{9}{5} = \frac{18}{5}\]
06

The point closest to (5, 2)

The point on the line \(y = 2x\) that is closest to the point (5, 2) is \((\frac{9}{5}, \frac{18}{5})\).

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In \(C[0,1],\) with inner product defined by \((3),\) compute (a) \(\left\langle e^{x}, e^{-x}\right\rangle\) (b) \(\langle x, \sin \pi x\rangle\) (c) \(\left\langle x^{2}, x^{3}\right\rangle\)

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