Find the x-value at which the tangent to the function of y= (4/e^2x-2) + 8x is parallel to the x-axis.

Step-by-step explanation:

The tangent line is parallel to the  

x

axis when the slope (hence  

d

y

d

x

) is zero and it is parallel to the  

y

axis when the slope (again,  

d

y

d

x

) goes to  

∞

or  

−

∞

Explanation:

We'll start by finding  

d

y

d

x

:

x

2

+

x

y

+

y

2

=

7

d

d

x

(

x

2

+

x

y

+

y

2

)

=

d

d

x

(

7

)

2

x

+

1

y

+

x

d

y

d

x

+

2

y

d

y

d

x

=

0

d

y

d

x

=

−

2

x

+

y

x

+

2

y

Now,  

d

y

d

x

=

0

when the nuimerator is  

0

, provided that this does not also make the denominator  

0

.

2

x

+

y

=

0

when  

y

=

−

2

x

We have now, two equations:

x

2

+

x

y

+

y

2

=

7

y

=

−

2

x

Solve (by substitution)

x

2

+

x

(

−

2

x

)

+

(

−

2

x

)

2

=

7

x

2

−

2

x

2

+

4

x

2

=

7

3

x

2

=

7

x

=

±

√

7

3

=

±

√

21

3

Using  

y

=

−

2

x

, we get

The tangent to the curve is horizontal at the two points:

(

√

21

3

,

−

2

√

21

3

)

and  

(

−

√

21

3

,

2

√

21

3

)

(Observe that these pair do not also make the denominator of  

d

y

d

x

equal to  

0

)

To find the points at which the tangent is vertical, make the denominator of  

d

y

d

x

equal tpo  

0

(without also making the numerator  

0

).

We could go through the solution, but the symmetry of the equation that we will get:

x

=

−

2

y

, so

y

=

±

√

21

3

and the points on the curve at which the tangent is vertical are:

(

−

2

√

21

3

,

√

21

3

)

and  

(

2

√

21

3

,

−

√

21

3

)

By the way. Because we do have the technology, here is the graph of this rotated ellipse: (Note that  

±

√

21

3

≈

±

1.528

which you can see on the graph.)

graph{x^2 + xy +y^2 =7 [-11.3, 11.2, -5.665, 5.585]}

Answer link

2

step-by-step explanation:

vvvvv

the answer should be a. snail