Let f : [a, b] → R be a differentiable function and let P be a partition of [a, b]. Show that L(P, f′) ≤ f(b) − f(a).

Question

 Let f : [a, b] → R be a differentiable function and let P be a partition of [a, b]. Show that L(P, f′) ≤ f(b) − f(a).

 

Hint: Use the Mean-Value Theorem.
(Recall that, if P = (x0,x1,...,xn), then L(P,f′) = Σmj∆xj, where mj = inf{f′(t) : t ∈
[xj−1,xj]}and∆xj =xj −xj−1 foreach1≤j≤n.)

Answer 1

Applying the MVT on the interval [x_j-1 , x_j], there is a number c_j in the interval (x_j-1 , x_j) such that

 

    f'(c_j) = [f(x_j) - f(x_j-1)] / Δx_j

 

Do this for each of the n intervals of the partition, P.

 

Then, ∑f'(c_j)Δx_j = ∑[(f(x_j) - f(x_j-1) / Δx_j)Δx_j]

 

                        = ∑(f(x_j) - f(x_j-1))

 

                        = [ f(x₁) - f(x₀)] + [f(x₂) - f(x₁)] + [f(x₃) - f(x₂)] + ... + [f(x_n) - f(x_n-1)]

 

                        = f(x_n) - f(x₀) = f(b) - f(a)

 

For each j from j = 1 to j = n, f'(c_j) ≥ f'(m_j)

 

So, ∑f'(m_j)Δx_j ≤ ∑f'(c_j)Δx_j = f(b) - f(a)

 

That is, L(f',P) ≤ f(b) - f(a)