What point on the line y 9x+4 is closest to the origin? let d be the distance between the two points. what is the objective function in terms of the x-coordinate? (type an expression.)

The objective function in terms of the x-coordinate is .

The point closest to the origin is .

Step-by-step explanation:

The formula for the distance from point (x, y) to the origin is

So, in our case, and the distance is

This is the objective function.

Next, we need to find the derivative of the function

Now, we set the derivative function equal to zero to find the critical points

We need to check that the value that we found is a minimum point for this we analyze the intervals of increase or decrease (First derivative test).

We can use a sign chart. In a sign chart, we pick a test value at each interval that is bounded by the critical points and check the derivative's sign on that value.

This is the sign chart for our function:

d(x) decreases before , increases after it, and is defined at . So d(x) has a relative minimum point at .

The point closest to the origin is

b: |12 – x| ≤ 0.25

step-by-step explanation:      

let x be the possible lengths in inches, of the ribbons jaime cut.

we have been given that jaime is cutting strips of ribbon for a project. he tries to cut each piece to be 12 inches long, but sometimes he cuts it too long or too short by as much as 0.25 inch.  

we can represent this information in an inequality as:

by the definition of absolute value function we will get,

upon combining our inequalities we will get,

therefore, the inequality   describes all the possible lengths (x), in inches, of the ribbons jaime cut.

step-by-step explanation: