Find
also the local maximum and the local minimum values, as the case may be:
(i) f (x) = x2
(ii) g(x) = x3 – 3x
(iii) h(x) = sin x + cos x, 0
2
x
π
<
<
(iv) f (x) = sin x – cos x, 0
2
<x
< π
(v) f (x) = x3 – 6x2 + 9x + 15
(vi)
2
( )
,
0
x2
g x
x
x
=
+
>
(vii)
2
1
( )
2
g x
x
=
+
(viii)
( )
1
, 0
1
=
−
<
<
f x
x
x
x
4 Prove that the following  functions do not have maxima or minima:
(i) f (x) = ex
(ii) g(x) = log x
(iii) h (x) = x3 + x2 + x +1
5 Find the absolute maximum value and the absolute minimum value of the following
functions in the given intervals:
(i) f (x) = x3, x ∈ [– 2, 2]
(ii) f (x) = sin x + cos x , x ∈ [0, π]
(iii) f (x) =
12
9
4
,
2,
2
2
x
x
x


−
∈ −




(iv)
2
( )
(
1)
3,
[ 3,1]
f x
x
x
=
−
+
∈ −
6 Find the maximum profit that a company can make, if the profit function is
given by
p(x) = 41 – 72x – 18x2
7