Sivakumar
MATH 171
Exercise Set 9
1. (i) Give an example of a function f and a point c such that c belongs to the domain of f ,
f 0 (c) = 0, but f does not have a local extremum at c.
(ii) Give an example of a function f and a point c such that f is continuous at c, f 0 (c) does
not exist, but f does not have a local extremum at c.
Note that in both cases above, c is a critical number for f .
The next two examples are slightly modified versions of questions from an old exam (1989) I gave
at the University of Alberta, my alma mater:
2. Suppose f is continuous on [a, b] and |f 0 (x)| ≤ 1 for every x in (a, b). Show that |f (v)−f (u)| ≤
|v − u| for every a ≤ u < v ≤ b.
sin x
, 0 < x ≤ π, and f (0) = 1. Let g(x) = x cos x − sin x, 0 ≤ x ≤ π.
x
(i) Observe that f and g are continuous on [0, π]. The continuity of f at 0 requires some
thought; everything else is quite clear.
3. Let f (x) =
(ii) Find g 0 (x), 0 < x < π, and use it to show that g is decreasing on [0, π].
(iii) Evaluate g(0) and use it, along with (ii), to show that g(x) < 0, for every 0 < x ≤ π.
g(x)
, 0 < x < π.
x2
(v) Use (iv) and (iii) to show that f is decreasing on [0, π].
sin a
a
(vi) Suppose 0 < a < b < π. Show that
> .
sin b
b
(iv) Show that f 0 (x) =
The function f considered in Example 3 above is called the sinc function. It plays a useful rôle in
Signal Analysis, in particular, the Theory of Sampling.
4. Show that the equation x101 + x51 + x − 1 = 0 has exactly one real root (that is, there exists
exactly one real number which satisfies the said equation). Do this in two steps: firstly, show
that there is at least one such root, and, secondly, that there cannot be more than one.
5. A number c is called a fixed point of a function f if f (c) = c. Suppose f is a differentiable
function and f 0 (x) < 1 for all x. Show that f has at most one fixed point. (That is, show that
f cannot have two or more fixed points.)
6. (from a high-school exam given in Madras, India; I happened to come upon this test some
years ago.) Find all possible values of the real number a such that the function
f (x) = x3 + ax2 + 48x + 12 is always increasing (that is, increasing on (−∞, ∞)).
7. Determine all possible values of the number b for which the function
f (x) = (b2 + b − 6) cos 2x + (b − 2)x + cos 1
has no critical number.
8. (i) Show that tan x ≥ x for every 0 ≤ x < π/2.
x
≤ tan−1 x ≤ x for every x ≥ 0.
(ii) Show that
1 + x2
1