Purchase Solution

# Real analysis

Not what you're looking for?

Based on the Rolle, Lagrange, Fermat and Taylor Theorems. ******************************************************
Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0.

Denote M = sup |f "(x)| where x is in [a,b]

and g:[a,b] --> R, g(x)=(1/2)(x-a)(b-x)

i) Prove that for all x in [a,b], there exists

Cx in (a,b) such that f(x)= - f "(Cx)g(x).

Cx is a constant dependent on x

##### Solution Summary

This is a proof regarding a twice differentiable function.

##### Solution Preview

Let x be a fixed point in the interval [a,b] and consider the function of y:
H(y)=g(x)f(y)-g(y)f(x). Then notice that H'(y)=g(x)f'(y)-g'(y)f(x) and H''(y)=g(x)f''(y)-g''(y)f(x)=g(x)f''(y)+f(x) (because g''=-1 as can easily be ...

##### Free BrainMass Quizzes

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.