Please help with integration: ...Show thatL(P, f′) ≤ f(b) − f(a).

[nocreativity]

Let f : [a, b] → R be a differentiable function and let P be a partition of [a, b]. Show thatL(P, f′) ≤ f(b) − f(a).
Hint: Use the Mean-Value Theorem.

Full Question:

 

[Dr.Peterson]

Let f : [a, b] → R be a differentiable function and let P be a partition of [a, b]. Show thatL(P, f′) ≤ f(b) − f(a).
Hint: Use the Mean-Value Theorem.

Full Question:

View attachment 10220

Thanks for including the whole problem, which tells us what is meant by L(P,f') in this author's notation.

But can you also tell us where you need help? If you haven't already done so, please read our guidelines and this summary.

Now, if you don't see where to start, start by writing out what the MVT says when applied to function f and interval [x_j-1, x_j]. You may see what to do, or we can discuss it further.

 

[nocreativity]

Could you expand in this furthur please

 

[stapel]

Could you expand in this furthur please

What did you get, when you wrote out what the MVT says in this situation, as suggested? Please be complete. Thank you!