Use implicit differentiation to find the points where the parabola defined by x2−2xy+y2+4x−8y+20=0 has horizontal and vertical tangent lines. the parabola has horizontal tangent lines at the point(s) . the parabola has vertical tangent lines at the point(s)

The parabola has a horizontal tangent line at the point (2,4)

The parabola has a vertical tangent line at the point (1,5)

Step-by-step explanation:

Ir order to perform the implicit differentiation, you have to differentiate with respect to x. Then, you have to use the conditions for horizontal and vertical tangent lines.

-To obtain horizontal tangent lines, the condition is:

(The slope is zero)

--To obtain vertical tangent lines, the condition is:

(The slope is undefined, therefore the denominator is set to zero)

Derivating respect to x:

=

Solving for dy/dx:

Applying the first conditon (slope is zero)

Solving for y (Adding 2x+4, dividing by 2)

y=x+2 (I)

Replacing (I) in the given equation:

Replacing it in (I)

y=(2)+2

y=4

Therefore, the parabola has a horizontal tangent line at the point (2,4)

Applying the second condition (slope is undefined where denominator is zero)

2y-2x-8=0

Adding 2x+8 both sides and dividing by 2:

y=x+4(II)

Replacing (II) in the given equation:

Replacing it in (II)

y=1+4

y=5

The parabola has vertical tangent lines at the point (1,5)

irrational, then, just means all the numbers that aren't rational. every integer is a rational number, since each integer n can be written in the form n/1. however, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers.

step-by-step explanation:

(2,6)

step-by-step explanation:

cause i just did this and got the answer

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