CBSE (Science) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution for Let F Defined on [0, 1] Be Twice Differentiable Such that | F"(X) | ≤ 1 for All X ∈ [0, 1]. If F(0) = F(1), Then Show that | F'(X) | < 1 for All X ∈ [ 0, 1] ? - CBSE (Science) Class 12 - Mathematics

Login
Create free account


      Forgot password?

Question

Let f defined on [0, 1] be twice differentiable such that | f"(x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?

Solution

If a function is continuous and differentiable and f(0) = f(1) in given domain x ∈ [0, 1],
then by Rolle's Theorem;
f'(x) = 0 for some x ∈ [0, 1]
Given: |f"(x)| ≤ 1
On integrating both sides we get,
|f'(x)| ≤ x
Now, within interval x ∈ [0, 1]
We get, |f' (x)| < 1.

  Is there an error in this question or solution?
Solution for question: Let F Defined on [0, 1] Be Twice Differentiable Such that | F"(X) | ≤ 1 for All X ∈ [0, 1]. If F(0) = F(1), Then Show that | F'(X) | < 1 for All X ∈ [ 0, 1] ? concept: Increasing and Decreasing Functions. For the courses CBSE (Science), CBSE (Commerce), CBSE (Arts), PUC Karnataka Science
S
View in app×