If f(x) = ∣(x2 − 10)(x2 + 1)∣, how many numbers in the interval 0 ≤ x ≤ 2.5 satisfy the conclusion of the mean value theorem?

none

one

two

three

  Two

Step-by-step explanation:

The mean value theorem tells you that a continuous, differentiable function will have at least one point between the endpoints of any interval where the slope of the function is equal to the average slope on the interval.

Here, f(x) is both continuous and differentiable on [0, 2.5], so the conclusion is guaranteed for at least one point. In the given interval, the curve f(x) crosses the line of average slope so has two points where a tangent has the average slope.

There are two values of x in the interval that satisfy the conclusion of the mean value theorem.

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:

sina-sinb

cosb+cosa

tana-tanb

cos0+cos x pi/3

20tan

step-by-step explanation: