1) Show that 2^x = 3x for some x element in (0,1)

2) Suppose f:[a,b]-->R is continuous and that f([a,b]) is a subset of Q (the rationals). Prove that f is constant on [a,b].

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- Apr 23rd 2009, 11:12 AMnoles2188properties of continuous functions
1) Show that 2^x = 3x for some x element in (0,1)

2) Suppose f:[a,b]-->R is continuous and that f([a,b]) is a subset of Q (the rationals). Prove that f is constant on [a,b]. - Apr 23rd 2009, 03:37 PMputnam120
1: Consider $\displaystyle f(x)=3x-2^x$ and then apply the intermediate value theorem to the interval [0,1].

2: Suppose that f is not constant then it takes on at least 2 different values. Apply the intermediate value theorem and the fact that between any 2 real numbers there is an irrational number.