The function $f$ is defined as $f(x)=2 x+3$ . $\newline$ What is the $x$ -coordinate of the point on the function's graph that is closest to the origin?

Full solution

Q. The function

f

is defined as

f(x)=2 x+3

\newline

What is the

x

-coordinate of the point on the function's graph that is closest to the origin?

Define Distance Formula: To find the point closest to the origin, we need to minimize the distance from the point $(x, f(x))$ to the origin $(0,0)$ . The distance formula is $D = \sqrt{((x-0)^2 + (f(x)-0)^2)}$ .
Substitute $f(x)$ : Substitute $f(x)$ with $2x+3$ to get the distance in terms of $x$ : $D = \sqrt{x^2 + (2x+3)^2}$ .
Simplify Distance Formula: Now we have $D = \sqrt{x^2 + (4x^2 + 12x + 9)}$ , which simplifies to $D = \sqrt{5x^2 + 12x + 9}$ .
Minimize Distance: To find the minimum distance, we can take the derivative of $D$ with respect to $x$ and set it to zero. But since $D$ is always positive, we can minimize $D^2$ for simplicity. So we'll minimize $5x^2 + 12x + 9$ .
Find Derivative: The derivative of $5x^2 + 12x + 9$ with respect to $x$ is $10x + 12$ .
Set Derivative Equal: Set the derivative equal to zero to find the critical point: $10x + 12 = 0$ .
Solve for x: Solve for x: $10x = -12$ .
Simplify Fraction: Divide by $10$ : $x = -\frac{12}{10}$ .
Simplify Fraction: Divide by $10$ : $x = -\frac{12}{10}$ . Simplify the fraction: $x = -\frac{6}{5}$ or $x = $-1$.$2$.