CURVE SKETCHING (GRAPHS) 0 115
Fig. 15.44
15.56 Sketch the graph of a function f(x) such that: /(0) = 0, /(2)=/(-2) = 1, /(0) = 0, f'(x)>0 for x>Q,
f'(x)<0 for x<0, /"(*)>0 for \x\<2, f"(x)<0 for |x|>2, lim^ f(x) = 2, \\m^f(x) = 2.
I See Fig. 15-45.
Fig. 15-45
Fig. 15-43
3*)
2
, 4(l-x)>>'
2 = (2-3;c)
2 . So, as *-»0, y'
2 ^l. Since 2yy'= x(2-3x),
as x-+Q\ /->!,
and, as x—»0~, _y'—>1. As *—»-°°, y—»-oo. Let us look for inflection points. Assume y" = Q.
Then, y'2 = l-3x, 4y2(l -3x) = x\2- 3x)2, 4x\l-x)(l-3x) = x2(2-3x)2, 4(1 - x)(l - 3x) = (23x)
2
, 4 - 16x + 12x
2 = 9^:
2 - I2x + 4, -16 + 12x = 9x - 12, 3* = 4, * = |. Hence, there are no inflection
points. See Fig. 15-43.
Sketch the graph of a function f(x) having the following properties: /(O) = 0, f(x) is continuous except at
x = 2, lim /(*) = +00, Hm /(x) = 0, lim f(x) — 3, f(x) is differentiable except at x=2 and jc=—1,
/'W>0
r "*if -1<^<2/7'W<0 if *<
+ -l or x>2, /"W<0 if x<0 and ^^-1, /"(jc)>0 if
x>0 and x^2.
I See Fig. 15-44.
15.55