The value of 'c of Lagranges Mean value
Theorem for f(x) = ^x when a=1, b = 4
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Answer:
Step-by-step explanation:
f(x)=
x
, where a=1,b=4
Checking conditions for Mean value theorem.
Condition 1:
f(x)=
x
is continuous at (1,4)
Since f(x) is polynomial.
It is continuous in (1,4)
Condition 2:
If f(x) is differentiable
f(x)=
x
f(x) is a polynomial and every polynomial function is differentiable
⇒ f(x) is differentiable at x∈[1,4]
Condition 3:
f(x)=
x
f
′
(x)=
2
1
x
1
f
′
(c)=
2
c
1
f(a)=f(1)
=
1
=1
f(b)=f(4)
=
4
=2
By Lagranges mean value theorem,
f
′
(c)=
b−a
f(b)−f(a)
2
c
1
=
4−1
2−1
2
c
1
=
3
1
c
=
2
3
∴ c=
4
9
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