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Answers
1 Yes. Either a local maximum or a local minimum for the function.
2 A differentiable function is continuous and a continuous function on a
closed interval is bounded and attains its bounds, from qns 7.31 and
7.34.
If f ( a )
f ( b ) is not a maximum then the maximum occurs on the open
interval. If f ( a )
f ( b ) is not a minimum then the minimum occurs on
the open interval. If f ( a )
f ( b ) is both a maximum and a minimum,
then the function is constant. Yes.
3 Yes, by qn 8.30.
4 If f ( a )
f ( b )
M m , there is a c with f ( c )
m .
f ( b )
m M , then there is a c with f ( c )
M .
If f ( a )
f ( b )
M m , then f ( x )
M m for all x , and so f
If f ( a )
( x )
0
from qn 8.6.
5 See figure 9.1.
6 Differentiable
Continuous.
At a local maximum or minimum f
( x )
0.
7
(i)
f is bounded and attains its bounds.
(iii)
when f ( a ) is not a maximum, there is a maximum f ( c ) with
a
c
b ; when f ( a ) is not a minimum there is a minimum f ( c )
with a
b ; and when f ( a ) is a maximum and a minimum f ( c )
is a maximum and a minimum.
c
(ii)
f
( c )
0.
8 See figure 9.2.
9 Apply Rolle's Theorem to the function
x a
x a
x /2 a
x /3 ... a
x /( n 1) on theintrval [0, 1].
10 Apply Rolle's Theorem to [ a , b ] and [ b , c ], to find f
0 with
a x b and f ( y ) 0 with b y c . Then apply Rolle's Theorem to
thefunction f on theintrval [ x , y ].
( x )
a
b
Figure9.1
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