Point on the graph of $y=\sqrt{4x+13}$ closest to $(5,0)$?

Question

Just did this question on an exam earlier today, I'm curious to see if I'm correct.

What point on the graph of $y=\sqrt{4x+13}$ is closest to $(5,0)$?

My answer: $(-1,3)$

Answer 1

Your method described in the comments seems correct. You start by noting that the distance between $(x,y(x))$ and $(5,0)$ is given by $$d(x) = \sqrt{(5 - x)^2 + (0 - y(x))^2} = \sqrt{(5 - x)^2 + 4x + 13} = \sqrt{x^2 - 6x + 38}. \tag{1}$$ You want to minimize $d(x)$, so indeed you want to look at the derivative: $$d'(x) = \frac{1}{2 \sqrt{x^2 - 6x + 38}} \cdot (2x - 6) = \frac{x - 3}{\sqrt{x^2 - 6x + 38}}$$ Solving $d'(x) = 0$ for $x$ we find $x = 3$, and this is indeed a global minimum; this could be proved by showing that $d''(3) > 0$.

So in the end we find a minimum at $(3, \sqrt{4\cdot3+13}) = (3,5)$, and this point has distance $d(x) = \sqrt{3^2 - 6\cdot 3 + 38} = \sqrt{29}$ from $(5,0)$.

---

In this case you can take a shortcut from $(1)$ to the solution as noted by Arthur, by noting that $$d(x)^2 = x^2 - 6x + 38 = (x - 3)^2 + 29 \geq 29$$ where the inequality follows from $(x-3)^2 \geq 0$, and equality holds only if $x - 3 = 0$, i.e. if $x = 3$.

Answer 2

Let $P = (x,\sqrt{4x+13})$ be a generic point of your function, and pose $Q=(5,0)$. We want to find the point $P$ such that the distance between $P$ and $Q$ is minimal. The distance is:

$$d(x) = \sqrt{(x-5)^2 + \left(\sqrt{4x+13} - 0 \right)^2} = $$ $$= \sqrt{(x-5)^2 + |4x+13|} $$

Minimize the distance is equivalent to minimize the square of the distance. That is:

$$f(x) = d^2(x) = (x-5)^2 + |4x+13| $$

Let's take the derivative of $f(x)$ (note that you have to take care about the absolute value!!!):

$$f'(x) = \left\{ \begin{array}{lr}2x - 10 + 4 & x \geq -\frac{13}{4}\\2x - 10 - 4 & x < -\frac{13}{4}\end{array}\right. = \left\{ \begin{array}{lr}2x - 6 & x \geq -\frac{13}{4}\\2x - 14 & x < -\frac{13}{4}\end{array}\right.$$

Now, we must pose $f'(x) = 0$, and then:

$$\left. \begin{array}{lr}2x-6 = 0 & x \geq -\frac{13}{4}\\2x-14 = 0 & x < -\frac{13}{4}\end{array}\right. \Rightarrow \left. \begin{array}{lr}x = 3 & x \geq -\frac{13}{4}\\x = 7 & x < -\frac{13}{4}\end{array}\right.$$

Cleary, the second solution is not admissible since $7 > -\frac{13}{4}$. Hence, the only solution is $x=3$ and then the point you are looking for is: $$P=(3, \sqrt{12+13}) = (3, 5)$$