We frequently must solve equations of the form f (x) = 0. When f is a continuous
Chapter 5, Problem 23(choose chapter or problem)
We frequently must solve equations of the form f (x) = 0. When f is a continuous function on [a, b] and f (a) and f (b) have opposite signs, the intermediate-value theorem guarantees that there exists at least one solution of the equation f (x) = 0 in [a, b]. (a) Explain in words why there exists exactly one solution in (a, b) if, in addition, f is differentiable in (a, b) and f _ (x) is either strictly positive or strictly negative throughout (a, b). (b) Use the result in (a) to show that x3 4x + 1 = 0 has exactly one solution in [1, 1].
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