1) Prove that if f\' < 0 on an interval, then f us decreasing on that interval.

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Question

1) Prove that if f' < 0 on an interval, then f us decreasing on that interval. (Use Mean Value Theorem)

2) Use the Mean value theorem to show that if f is continuous on the interval [x, x+h] and is differentiable on (x, x+h), then

f (x +h) - f (x) = f' (x + theta(h))h for some number (theta) between 0 and 1.

3) Let f(x) = (x^2 + 2x + 2) when -1/2 is less than or equal to x which is less than 0, and (x^2 - 2x + 2) when 0 is less than or equal to x which is less than or equal to 2.

a) show that f'(0) does not exist

b) Find the critical numbers of f.

c) Use the first derivative test to find the local max and local min values of f.

d) show that f is continuous at 0.

e) explain why f has both an absolute max and an absolute min.

f) Find the abs max and abs min values of f on the given interval.

Explanation / Answer

[f(x+h) - f(x)]/h <0

ie f(x+h)<f(x)

hence every value in between ie[f(x+h) +f(x)]/2 is less than f(x+h)

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1) Prove that if f\' < 0 on an interval, then f us decreasing on that interval.

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Explanation / Answer

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