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# 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 (Commerce) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### 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.

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Solution 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.
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