Final answer:
To find the point (x,y) at which the function f(x)=2−4x is closest to the origin, we need to minimize the distance between the origin and any point on the function. We can use the distance formula d = sqrt(x^2 + y^2). By substituting the function f(x) into this expression, we get a quadratic equation. We can solve this equation to find the value of x that minimizes the distance.
Step-by-step explanation:
To find the point (x,y) at which the function f(x)=2−4x is closest to the origin, we need to minimize the distance between the origin and any point on the function. We can use the distance formula d = sqrt(x^2 + y^2). Since we want to minimize the distance, we can minimize the square of the distance, which is x^2 + y^2. By substituting the function f(x) into this expression, we get a quadratic equation. We can then solve this equation to find the value of x that minimizes the distance.
Let's rearrange the function f(x) = 2−4x into the form y = mx + b, where m is the slope and b is the y-intercept. We have y = -4x + 2. The slope of this line is -4. Since we want to find the point closest to the origin, we want a line perpendicular to this line that passes through the origin. The slope of a line perpendicular to a line with slope m is the negative reciprocal of m. So the slope of the line perpendicular to y = -4x + 2 is 1/4.
Now we can use this slope to find the equation of the line that passes through the origin. The equation of a line with slope m and passing through the point (0,0) is y = mx. Substituting our slope of 1/4, we get y = (1/4)x. Now we have two equations: y = -4x + 2 and y = (1/4)x. We can solve these two equations to find the point (x,y) at which the function f(x)=2−4x is closest to the origin.