Example 14
Verify that f (x) = e^{x} sin
x satisfies the hypothesis for the Rolle’s theorem on [0, p]
and find number c described in the theorem.
Solution : f (x) = ex
sin x is continuous on [0, p]
and differentiable in (0, p)
Also, f (0) = f (p) = 0
\ All
the conditions of the Rolle’s theorem are satisfied.
\ There
exists a x = c Î(0,
p) such that f ' (c) = 0
Now f '(x) =
e^{x} cos x + e^{x}
sin x
= e^{x}
(cos x + sin x)
\
f '(c) = e^{c} (sin c + cos c)
\
0 = e^{c} (sin
c + cos c)
OR sin c + cos
c = 0 Þ tan
c = 1 Þ c
=
Î(0,
p).
Hence the Rolle's theorem is verified.
Example 15
Verify Meanvalue theorem for x  x^{3}
in (2 , 1)
Solution : f (x) = x  x^{3}
\ f
'(x) = 1  3x^{2}
\ f
(x) is continuous on [ 2, 1] and derivable in (2, 1) and
f (2) = 2 + 8 = 6 and f (1)
= 1  1 = 0
All conditions of L.M.V. theorem are satisfied.
\
There exists one x = c in (2 , 1) such that
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