Real Analysis : Young's Inequality

Note: * = infinite Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*). Then define F(x) = the integral from 0 to x of f and G(x) = the integral from 0 to x of f^-1 for all x>=0 (a) Prove Young's Inequality: ab <= F(a) + G(b) for all a >= 0 and b >= 0 (b) Now use Young's Inequality with f(x) = x^(p-1) for all x>=0, and p>1 fixed, to prove that if the number q is chosen to have the property that 1/p + 1/q = 1, then ab <= a^p/p + b^q/q for a >= 0 and b >= 0.

Differential Equations

Define: f(x) = (x^2)sin(1/x)+x if x doesn't equal 0 f(x) = 0 if x=0 Prove that the function f:R-> R is differentiable and that f'(0)=1. Also prove that there is no neighbourhood I of 0 such that the function f:I->R is increasing.

Taylor polynomials

Prove that: 1 + x/2 - (x^2)/8 < squareroot(1+x) < 1 + x/2 if x>0 In particular, show that 1.375 < squareroot(2) < 1.5

Approximation with Taylor polynomials

note:% just a symbol Let I be an open interval containing the point x.(x not), and suppose that the function f:I->R has a continuous third derivative with f'''(x)>0 for all x in I. Prove that if x.+h is in I, there is a unique number % = %(h) in the interval (0,1) such that f(x.+h) = f(x.) + f'(x.)h + f"(x.+%h)(h^2)/2 and prove that limh->0 of %(h) = 1/3

Approximation with Taylor polynomials

Suppose that the function F:R->R has derivatives of all orders and that: F"(x) - F'(x) - F(x) = 0 for all x F(0)=1 and F'(0)=1 Find a recursive formula for the coefficients of the nth Taylor polynomial for F:R->R at x=0. Show that the Taylor expansion converges at every point.