MATHEMATICS 120 PROBLEM SET 3
Due September 25, 2002
For full credit, please show all work.
1. (5 marks) The function f (x) is defined by

0 if − 2 ≤ x ≤ −1,



 ax2 + bx if − 1 < x < 2,
f (x) =

c if x = 2,



2x if 2 < x ≤ 5.
Find all values a, b, c such that f (x) is continuous on [−2, 5].
2. (5 marks) Use the intermediate value theorem to prove that the equation x3 −7x+2 = 0
has three solutions in the interval [−3, 3].
3. (5 marks) Give an example of a function f (x), defined on an open interval, such that
f (x) has a maximum value but does not have a minimum value. Also give an example of a
function g(x), defined for all x, such that g(x) has both a maximum value and a minimum
value. (You can give a formula or draw a graph – either is acceptable.)
4. (5 marks) Assume that the following is true for some function f (x), defined for all x:
for every > 0 there is a δ > 0 such that |f (x) −L| < for some x such that 0 < |x −a| < δ
(read this carefully!). Does this imply that limx→a f (x) = L? Explain why or why not.
Please read Sections 1.4–1.5 of the textbook and make sure that you can solve the practice
problems at the end of each section.
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