‫كلية العلوم‬
College of Sciences
‫قسم العلوم الرايضية‬
Department of Mathematical Sciences
Work sheet 2-Math 101
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1) In each of the following figure, does the given function f satisfy the
hypotheses of Roll’s theorem on the interval [-3,3]? If so, find a
value of c that satisfies the conclusion of Roll’s theorem on that
interval.
A)
B)
C)
1
1
4
3
2) Given the functions 𝐼) 𝑓(𝑥) = 𝑥 4 + 𝑥 3 − 3𝑥 2 + 5
𝐼𝐼) 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 − √3,
find the following:
A) The intervals where f(x) is increasing or decreasing
B) The intervals where f(x) is concave up or down
C) The inflection points (if any)
D) The relative extrema (maximum or minimum)
3) Find all critical point of the following function. Then check whether
the function has an extremum at these critical points.
𝐼) 𝑓(𝑥) = √𝑥 2 + 1
2
𝐼𝐼) 𝑓(𝑥) = (3𝑥 + 1)3
5) 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑤ℎ𝑒𝑡ℎ𝑒𝑟 𝑎𝑙𝑙 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑒𝑠 𝑜𝑓 𝑀𝑒𝑎𝑛−𝑉𝑎𝑙𝑢𝑒−
𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝑎𝑟𝑒 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑡𝑎𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 .𝐼𝑓 𝑠𝑜,𝑓𝑖𝑛𝑑 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠
𝑜𝑓 𝑐 𝑔𝑢𝑎𝑟𝑎𝑛𝑡𝑒𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑜𝑛𝑐𝑙𝑢𝑠𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑇ℎ𝑒𝑜𝑟𝑒𝑚
𝑓(𝑥) = 2𝑥 2 − 3𝑥 + 1,
[0,2]