Question

Locate the absolute extrema of the function (if any exist) over each interval.

f (x) = x^2 - 2x

(a) [-1, 2]        (b) (1, 3 ]             (c) (0, 2)            (d) [ 1, 4)

Answer 1

Step 1:

(a)

The function and the interval is .

Differentiate the function with respect to

Find the critical numbers of , by setting .

Critical number is .

Step 2:

Find the values of at this critical number .

Find the values of at the end points of the interval .

Compare the three values of to find absolute extrema of the function.

Absolute maximum value is

Absolute minimum value is

Solution:

Absolute maximum value is

Absolute minimum value is

Answer 2

Step 1:

(b)

The function and the interval is .

Find the values of at the end points of the interval .

Since open interval is there at 1, exclude that point.

Absolute maximum value is

Step 2:

(c)

The function and the interval is .

Find the value of the function at critical number .

Since open interval is there at 2, exclude that point.

Absolute minimum value is .

Step 3:

(d)

The function and the interval is .

Find the value of the function at critical number .

Since open interval is there at 4, exclude that point.

Absolute minimum value is .

Solution:

(b)

Absolute maximum value is

(c)

Absolute minimum value is .

(d)

Absolute minimum value is .