# Show the relationship between the supremum and infimum of f^2 and |f|

Suppose f: [a,b] $$\to$$ $$\mathbb{R}$$ and B satisfy |f(x)| $$\le$$ B for every x $$\epsilon$$ [a,b].

Show that if P = {x$$_{0}$$,...,x$$_{n}$$} is a partition of [a,b], then

M(f$$^{2}$$,[x$$_{i-1}$$,x$$_{i}$$]) - m(f$$^{2}$$,[x$$_{i-1}$$,x$$_{i}$$]) $$\le$$ 2B(M(f,[x$$_{i-1}$$,x$$_{i}$$]) - m(f,[x$$_{i-1}$$,x$$_{i}$$]))

for every 1 $$\le$$ i $$\le$$ n.

We are given a hint, namely that

|f(x)$$^{2}$$ - f(y)$$^{2}$$| = |f(x) - f(y)||f(x) + f(y)|.

And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.

• What are the functions $M$ and $m$? – Sean Roberson Nov 22 '18 at 2:51
• Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets. – user610107 Nov 22 '18 at 2:55
• The result is obvious from the hint and triangle inequality $|f(x) +f(y) |\leq 2B$ and $|f(x) - f(y) |\leq M_f-m_f$ – Paramanand Singh Nov 22 '18 at 4:18