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Unformatted text preview: numbers f ( x ) 6 = 1, then f has at most one ﬁxed point. 5. Suppose the derivative of a function f is given by f ( x ) = ( x + 1) 2 ( x-3) 5 ( x-6) 4 . On what interval is f increasing? 6. Suppose f is diﬀerentiable on an interval I and f ( x ) > 0 for all numbers x ∈ I , except for a single number. Prove that f is increasing on the entire interval. 7. Consider the function f deﬁned by f ( x ) = ( if x = 0 x 4 sin 1 x if x 6 = 0 Show that 0 is a critical point. Can you use the ﬁrst derivative test to ﬁgure out whether 0 is a local extrema? Why? 8. Use the mean-value theorem to deduce that ± ± sin x-sin y ± ± < ± ± x-y ± ± . 1...
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- Spring '10
- Electrical Engineering