
7-16 Calculus – Differentiation and Integration
Then the distance between (x, y) and (3, 0) is given by
S x y x x y
x x x
x x
= − + − = + − +
= + − + −
= − +
( ) ( )
.
3 0 9 6
9 6 5
7 14
2 2 2 2
2
2
Now, we need to minimise the distance S.
We have
dS
dx
x
x x
=
−
− +
2 7
2 7 14
2
.
Thus,
dS
dx
x x
x
x
=
⇒
− +
− =
⇒ =
0
1
2 7 14
2 7 0
7
2
2
( )
.
Thus,
y =
3
2
. And clearly, S is minimum at x = 7/2.
Therefore, the nearest point to the point (3, 0) on the graph of
5- x is
7
2
3
2
, .
And the shortest possible distance is
S =
7
.
Example 7.12 Find the equations of the tangent lines with maximum and minimum slopes of all the
tangent lines to the curve
y
x
=
+
3
3
2
.
Solution: The slope of ...