##
Techniques of Differentiation - Exercise 4

**Exercise 4.** Find the points on the graph of
*y* = *x*^{3/2} -
*x*^{1/2} at which the tangent line is parallel to the line *y*+2*x*
= 1. Also find the points on the same graph at which the
tangent line is perpendicular to the line *y*-*x* = 3.

**Answer.** First let us find the points on the graph at which
the tangent line is parallel to the line *y*+2*x* = 1. For that,
we need the slope of any tangent line which is given by the
derivative

We know that two lines are parallel if and only if they have the
same slope. Since the slope of the line *y*+2*x* = 1 is -2, we
then are left to solve

We rewrite this equation to get

which is a quadratic equation in .
Therefore we must
have

Since
has to be non-negative, we have to discard the
negative solution and are thus left with

or equivalently

So there is only one point on the graph at which the tangent line
is parallel to the line *y*+2*x* = 1.
Next we look for the points on the graph at which the tangent
line is perpendicular to the line *y*-*x* = 3. In this case, the
slope of tangent line should be

As before we must solve the equation

We rewrite this equation to get

Therefore

which implies
or
equivalently

This is the only point on the graph at which the tangent line is
perpendicular to the line *y*-*x* = 3.

**
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