Rolle's Theorem and Mean Value Theorem

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Rolle's Theorem

Let f be a real valued function in [a,b] such that

• f is continuous in [a,b].

• f is differentiable in (a,b).

Geometrical meaning

Let A (a,f (a)) and B (b,f (b)) be two points on the graph of f (x) such that f(a) = f(b), then  $ c Î (a, b) such that the tangent at P(c, f(c)) is parallel to x - axis.

Note 1:

We cannot obtain c if any one of the conditions of Rolle's theorem are not satisfied.

Note 2:

The value of c need not be unique.

Example:

Verify Rolle's theorem for the function

f (x) = x² - 8x + 12 on (2, 6)

Since a polynomial function is continuous and differentiable everywhere f (x) is differentiable and continuous (i) and (ii) conditions of Rolle's theorem is satisfied.

f (2) = 2² - 8 (2) + 12 = 0

f (6) = 36 - 48 + 12 = 0

Therefore (iii) condition is satisfied.

Rolle's theorem is applicable for the given function f (x).

\ There must exist c (2, 6) such that f '(c) = 0

f '(x) = 2x - 8

Rolle's theorem is verified.

Working Rule for Verifying Rolle's Theorem

Let f (x) be a function defined on [a, b].

Step 1:

Show that the function is continuous in the given interval. Some known standard functions which are continuous, can be mentioned directly.

Step 2:

Differentiate f (x) and examine if f '(x) is defined at every point in the open interval (a, b).

Step 3:

Check if f (a) = f (b)

If all the above condition are satisfied, then Rolle's theorem is applicable else the Rolle's theorem is not applicable.

If Rolle's theorem is applicable, solve f '(c) = 0. Show that one of these roots lie in the open interval (a, b).