ADDIS ABABA UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 2021 WORKSHEET 4
1.
Explain why the Mean Value Theorem (MVT) doesn’t apply for
a)
f(x) = | 2x – 1 | - 3 on [ -1, 2]
x2 1
b)
f(x) =
on [ 0,2]
c)
f(x) = x2/3 on [ -1,1,]
x 1
2.
Let p(x) = an xn + an-1 xn-1 + … + a1 x + a0. Suppose p(a) = p(b) = 0 and p(x)  0
x  (a, b). Prove that
a)
P(x) is strictly monotonic on [ a, b]
b)
There is at most one number r  (a, b) such that P(r) = 0
3.
Give the number of real roots and locate each root between successive integers for
the equation.
a)
2x3 – 9x2 + 1 = 0
b)
3x5 + 5x + 1 = 0
4.
Using the MVT, show that
a)
If | f (x) |  1 x in some interval I, then |f(x1) – f(x2) |  x1 – x2| x1,x2I
b)
| sin x|  | x | x  
5.
Suppose f is differentiable on [a, b]. Prove that if the minimum of f on [ a, b] is a,
then f (a)  0, and if it is at b, then f (b)  0.
6.
For each of the following, find the critical points, the local extreme points and the
intervals on which the given function is strictly monotonic.
f(x) = -x4 + 3x3 – 2x2
x 2  6x
f(x) =
( x  1) 2
b)
f(x) = (x – 2)2/3
d)
f(x) = | x2 + x – 2 |
e)
f(x) = ex + e-x
f)
f(x) = | x – 2 | + | x – 4|
a)
Suppose f is differentiable on an open interval I. Let a, b  I such that
f (a) < 0 and f (b) > 0 with a < b. Show that f (x) = 0 for some x in
(a, b). Hint consider the minimum of f on [ a, b].
c)
If f  exists and f (x)  0, for each x in an interval I, then show that f is
strictly monotonic on I.
a)
c)
7.
8.
Find the local extreme values, intervals of monotonicity, intervals of concavity,
inflection points and sketch the graph of:
x3
4
3
a)
f(x) = 3x – 4x
b)
f(x) =
1 x2
c)
f(x) = x +
x
d)
f(x) = x 1  x 2
9.
Of all the triangles that pass through the point (1, 1) and have sides lying on the
coordinate axes, one has the smallest area. Determine the lengths of its sides.
10.
Find the dimension of the right circular cylinder of largest volume that can be
inscribed in a sphere of radius 10 units.
11.
A leader is to reach over a fence 8 fit high to a wall 1 foot behind the fence. What
is the length of the shortest ladder that can be used?
12.
A light is affixed to the top of a 12 ft-tall lamppost. A 6 ft-tall man walks away
from the foot of the lamp post at a rate of 5 ft/sec. How fast is the length of his
shadow increasing when he is 5 ft away?
13.
Two roads intersect at right angle. A car traveling 80 km/hr reaches the
intersection half an hour before a bus that is traveling on the other road at 60
km/hr. How fast is the distance between the car and the bus increasing 1 hr after
the bus reaches the intersection?
14.
Find the point on the graph of the equation y = 4x which is nearest to the point
(1,2).
15.
Show the greatest area of any rectangle inscribed in a triangle is half that of the
triangle.