F(X)={ax-b,x->2
{x^2-3,x<2
IS differentiable at x= 2
find a,b
Answers
Answered by
1
Answer:
b
=
15
2
Explanation:
As
f
(
x
)
is continuous at
x
=
2
, we have
lim
x
→
2
−
f
(
x
)
=
lim
x
→
2
+
f
(
x
)
⇒
a
(
2
4
)
+
5
(
2
)
=
b
(
2
2
)
−
3
(
2
)
⇒
16
a
+
10
=
4
b
−
6
⇒
a
=
1
4
b
−
1
As
f
(
x
)
is differentiable at
x
=
2
, the limit
f
'
(
2
)
=
lim
x
→
2
f
(
x
)
−
f
(
2
)
x
−
2
must exist. We can tell what the one sided limits will evaluate to by calculating the derivatives of the components of the piecewise defined functions on either side of
2
.
lim
x
→
2
−
f
(
x
)
−
f
(
2
)
x
−
2
=
lim
x
→
2
+
f
(
x
)
−
f
(
2
)
x
−
2
⇒
4
a
(
2
3
)
+
5
=
2
b
(
2
)
−
3
⇒
32
a
+
5
=
4
b
−
3
Substituting in
a
=
1
4
b
−
1
, we have
32
(
1
4
b
−
1
)
+
5
=
4
b
−
3
⇒
8
b
−
27
=
4
b
−
3
⇒
4
b
=
30
∴
b
=
15
2
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