Question

Find the point where the line tangent to the graph of f(x)=(2x^3-12x^2+18x+7) is horizontal.

Answer 1

So I find the derivative (6x^2-24x+18) and I think I must set it equal to zero. But then I fumble.

Answer 2

You're on the right track.

Answer 3

the line tangent to f(x) is horizontal if it has slope zero. The derivative represents the slope, so you correctly set it to zero. What does that represent?

Answer 4

Asymptote?

Answer 5

f ' (x) = 0 >>> the x that satisfies this is the x value where the slope of f(x) is zero, meaning it's the point where the tangent (with same slope, def'n of tangent) is horizontal. But you need the point, not just the x value. So you need to put the x value that creates f'(x) = 0 back into f(x) to get the ordered pair for the solution

Answer 6

Does that make any sense? I can draw a made-up sketch that might help...

Answer 7

What x value do you mean that creates the derivative of x?

Answer 8

So, your first step you found derivative, which is the expression of slope everywhere on f(x). By setting to zero in step 2 and solving for x, you find the x value where the derivative is zero, which is equivalent to saying "the x value where the original function's slope is horizontal or zero". So put that actual x solution back into f(x) to get the rest of the ordered pair (x,y) which is the spot on f(x) where the tangent line touches AND is horizontal

Answer 9

Ah, so I must plug that value back into the original equation then?

Answer 10

yes, exactly :)

Answer 11

When you get that point, the point will be the place where the tangent line is horizontal.

Answer 12

Real world example, quickly...

Think of a roller coaster. Slope changes as you go up and down, so f ' (x) is not constant but is a function of x. By setting f ' (x) = 0, you declare that you want points where the roller coaster is going flat, even for just a point... top of a big hill just as it tips over the top, or bottom of a big hill, right as it turns to go back up.

Answer 13

By putting that x value back in the roller coaster "equation", you not only find out how far horizontally you are when it's flat, you find out the f(x) "height" of the coaster at that spot.

Answer 14

Ah, good analogy. Well thanks, Jake. I think I should be able to solve it now.

Answer 15

Good :) Glad to help... remember the roller coaster when you do 2nd derivatives... useful analogy to keep you from getting lost in the calculations :)

Answer 16

Cool, will do. Take care :)