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I am doing some exercises for my real analysis class and I ran into one that has been bugging me all day. I am using the book Elementary Analysis Theory of Calculus by Kenneth Ross. The problem is in Section 33 and is exercise 33.6 and it reads the following: Prove that for any subset S of [a,b], M(|f|,S)-m(|f|,S)<=M(f,S)-m(f,S). Hint: For x,y in S |f(x)|-|f(y)|<=|f(x)-f(y)|<=M(f,s)-m(f,s). Note that this book might have different notation than what some of you are used to but just to be clear, M(f,S)=sup{f(x):x in S} and m(f,S)=inf{f(x):x in S}. Any help will be appreciated. Thanks..

Hello. I'm studying for my analysis final and I have no idea how to do this problem. It says the following: Suppose f(x)=x^3+3x^2-2x-1 is continuous everywhere on R. Use the intermediate value theorem to show that f(x) has three distinct real roots. Hint: Evaluate f(x) at x=-1,0, an 1 and make use of the limit behavior of f(x) as x approaches plus or minus infinity. I calculated the values and got that f(-1)=3, f(0)=-1, and f(1)=1. Also, as x approaches negative infinity, f(x) approaches negative infinity and as x approaches positive infinity, f(x) approaches positive infinity. When I graph it, I can see that there are three real roots but I just don't know how to go about the proof. Any suggestions or help? Thank you.