Calculus: Analytic Geometry and Calculus, with Vectors

298 Functions, graphs, and numbers

2 Letting f(x) = axe + bx + c where a > 0,ll show that

f (x) = 2a (x -{ 2a).

Tell why f is decreasing when x < -b/2a and increasing when x > -b/2a.

Show that f has a global minimum at the point

b

2a'

b2 - 4acll

4a /

3 Letting f (x) = axe + bx + c where a < 0, show that f is increasing when

x < -b/2a, that f is decreasing when x > -b/2a, and that f has a global maxi-

mum at -b/2a.

4 Letting f (x) = 1/(1 + x2), show that f is increasing when x < 0, is decreas-

ing when x > 0, and hence has a global maximum when x = 0.

5 Show that the function f for which

3

f(x) = 1 +x2

is everywhere increasing and hence has no extrema.
6 Show that
x
x22-1
is decreasing except when x = ±1.

(^7) Find all trends and extrema of the function f for which
AX) =1 + x2
and sketch a graph of y = f(x). Hint: After calculating f'(x), put the result in
the form
f,(x) _
(1 + x)(1 - x)
(1 + x2)2
and, after observing that f'(-1) = 0 and f'(1) = 0, find the sign of f'(x) over
each of the intervals x < -1, -1 < x < 1, and x > 1. Then find f(-1) and
f(1) and make efficient use of this information. Find f'(0) and make the graph
have the correct slope at the origin.
(^8) Supposing first that x > 0, find the trends and extrema of the function
f for which
f(x) = x + z
and sketch the graph of y = f(x). Then let x < 0 and repeat the process without
use of symmetry, but use symmetry to check the results that are obtained.
(^9) This problem requires us to think about making tanks from rectangular
pieces of sheet iron. Starting with a rectangle 15 units wide and 24 units long,
we cut equal squares from the four corners and fold up the flaps to form a tank.