Derivative Theorems Part II

I.  Quiz

II. Homework

III.  Finding Roots

Example:  Show that f(x) = x⁴ + 10x³ + 3x + 1

Show that f(x) has a root between -1 and 1

Solution:  f(-1) = 1 - 10 - 3 + 1 = -11 and f(1) = 1 + 10 + 3 + 1 = 15

since 0 is in [-11,15]  and f(x) is continuous, IVT tells us that there is a c in [-1,1] with f(c) = 0 .

Programming a computer to locate a root:

1)  Divide the x-range into 2 regions

2)  Evaluate the function at the endpoints of each region.

3)  For each region if the signs of the endpoints differ, then use this region as the next range and go back to step 1.  

Try this with cos x - x

Putting the theorems together

Show that if f(x) = x³ - 5x² + 2x + 2  

then f'(c) has a root.

Hint: consider f(1), f(2) and f(10) and use the IVT and then Roll's theorem.