Question

Verify that the intermediate value theorem applies to the indicated interval and find the value of C guaranteed by the theorem.

f (x) = x³ - x² + x - 2,   [0, 3],   f (c) = 4

Answer 1

Step 1:

The function is and the interval is .

Since the function is a polynomial, it is continuous on the interval .

At  , .

At  ,

               

It follows that and .

Hence , apply the intermediate theorem to state that there must be some in such that .

Step 2:

Solve the equation .

Solve .

Compare with standard form of quadratic equation .

.

Solution  .

.

Since above two roots are imaginary consider the real root .

Solution:

The value of .