Finding the tangent line to the graph of y=(x^2-1)^2 that is perpendicular to the line x+5=0?

Question

This is a question for some calculus h/w that i have. here is the exact question:
Two tangents to the graph of y=(x^2-1)^2 are perpendicular to the line x+5=0. Determine the equations of the tangents.

the answer at the back of the book says y=0 and y=1, which, to me, dosn't even make sense... i have a quix on it on monday so i really need to know how to do it :) ty!

Answer 1

y = (x^2-1)^2 = x^4 - 2x^2 + 1

dy/dx = 4x^3 - 4x = 4x(x+1)(x-1)
= 0 when x=0, x = 1 or x = -1
x = 1 y = 0 (same as x=-1)
x = 0 y = 1
The line x+5 = 0 is the line x=-5 i.e vertical - this is why you need the derivative to be equal to 0 i.e. horizontal and horizontal lines are give by the equation y = constant

Answer 2

The line x + 5 = 0 is a vertical line. So you are looking for tangent lines perpendicular to this. They will be horizontal lines. Take the derivative of the function and set it equal to zero. The points where the derivative is zero will have horizontal tangents.

y = (x² - 1)²

dy/dx = 2(x² - 1)(2x) = 4x(x² - 1) = 4x(x - 1)(x + 1) = 0
x = -1, 0, 1

Solving for y at each of these points we have:
(-1,0), (0,1), and (1,0)

So the tangents to the curve at these points are:

y = 0 and y = 1.

Note that the line y = 0 is tangent to two of the points.

Answer 3

If attractiveness were time, you'd be eternity. If I were a provide up gentle, i might want to turn pink whenever you surpassed through, merely so i might want to stare at you slightly longer. Babe, your attractiveness makes the morning solar appear as if the boring glimmer of the moon. you'll extra efficient direct that attractiveness another position, you'll set the carpet on fireplace. If I received a nickel for everytime I suggested someone as pleasing as you, i might want to have 5 cents. For a second i presumed I had died and lengthy gone to heaven. Now I see that i'm very a lot alive, and heaven has been presented to me. Sorry, yet you owe me a drink. [Why?] because when I appeared at you, I dropped mine. You seem chilly. opt to apply me as a blanket? am i able to have guidelines? [To the position?] on your heart. Do you've a BandAid? I merely scraped my knee falling for you.

Answer 4

If abc is equal to def. then xwy :/ e=MC2. You cant go wrong