Student number Name [SURNAME(S), Givenname(s)] MATH 100, Section 110 (CSP)

Student number
Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Weeks 9 and 10: Marked Homework Assignment
Due: Thu 2010 Nov 18 14:00
1. Prove that 1 + 2x + x3 + 4x5 has exactly one real root.
2. Suppose f (x) is continuous for 1 ≤ x ≤ 2 and f 0 (x) exists for 1 < x < 2.
(a) If f (1) < f (2), prove that f 0 must be positive at some point.
(b) If f (1) = 5 and |f 0 (x)| ≤ 4 for all 1 < x < 2, prove that f (2) is between 1 and 9.
Is it possible for f (2) to actually equal 1 or 9? Justify your answer.
3. Let L be the tangent line to the curve y = 1 + ex at a point P (a, b), where the point
of tangency P is chosen so that the tangent line L passes through the origin O(0, 0).
(a) Find an equation that is satisfied by a, the x-coordinate of the point of tangency
(b) Prove that the equation in part (a) has exactly one real solution.
4. Consider the set of all functions f (x) defined on the closed interval [0, 1] that satisfy
the following three properties:
i) f is continuous in [0, 1];
ii) f is differentiable in (0, 1);
iii) f (0) = 0 and f (1) = 1.
Find a specific function f0 defined on [0, 1] that has these three properties, and in
addition the maximum of |f00 (x)| on (0, 1) is as small as possible. Justify your answer.
5. For each of the following functions, determine i) all critical numbers, ii) intervals where
the function is increasing or decreasing, iii) local extrema, iv) intervals where the graph
is concave upward or downward, v) the location of any inflection points, and vi) the
behaviour at infinity (if applicable). Then sketch the graph of the function, showing
all the information i)–vi) determined above, and also detail near any critical numbers:
(a) f (x) = 8x5 − 5x4 − 20x3
(b) g(x) = x1/3 − x2/3
(c) h(t) = t2/3 (t − 2)2