The function $f$ is defined as $f(x)=2 \sqrt{9-x}$ . $\newline$ 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 \sqrt{9-x}

\newline

What is the

x

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

Define Function: The function is $f(x) = 2\sqrt{9 - x}$ . To find the $x$ -coordinate of the point closest to the origin, we need to minimize the distance from the origin to the point on the graph.
Distance Formula: The distance from the origin to any point $(x, f(x))$ on the graph is given by the distance formula: $D = \sqrt{x^2 + f(x)^2}$ .
Substitute and Simplify: Substitute $f(x)$ into the distance formula: $D = \sqrt{x^2 + (2\sqrt{9 - x})^2}$ .
Minimize Distance: Simplify the distance formula: $D = \sqrt{x^2 + 4(9 - x)}$ .
Expand Expression: To minimize $D$ , we need to minimize the expression under the square root since the square root function is increasing. So we minimize $x^2 + 4(9 - x)$ .
Complete the Square: Expand the expression: $x^2 + 4(9 - x) = x^2 + 36 - 4x$ .
Complete the Square: Expand the expression: $x^2 + 4(9 - x) = x^2 + 36 - 4x$ .To find the minimum, we can complete the square or take the derivative and set it to zero. Since we're looking for a simple solution, let's complete the square.
Complete the Square: Expand the expression: $x^2 + 4(9 - x) = x^2 + 36 - 4x$ .To find the minimum, we can complete the square or take the derivative and set it to zero. Since we're looking for a simple solution, let's complete the square.Rewrite the expression as a perfect square: $x^2 - 4x + 36 = (x - 2)^2 + 32$ .