Vertical tangent lines
b. Does the curve have any hori...

Question

Vertical tangent lines

b. Does the curve have any horizontal tangent lines? Explain.

Accepted Answer

Welcome back, everyone. Are there any horizontal tangent lines on the curve defined by -2 X + 4 Y cubed plus Y equals 9? Justify your answer. We're given 4 answer choices. A says yes. Since the first derivative is equal to 0 at X equals -3 and 1, there are two horizontal tangent lines. B says yes, since the first derivative is equal to 0 at X equals negative. 3 and X equals -2. There are two horizontal tangent lines. C says yes, since the first derivative is equal to 0 at X equals 1, there is one horizontal tangent line and D says none because the first derivative will never be zero. So for this problem, we're going to apply the concept of implicit differentiation. Let's take the left hand side, -2 X. Plus 4 Y cubed plus Y, we're going to take the derivative of this expression and we're going to set it equal to the derivative of 9. For the left hand side we can rewrite it as -2, the derivative of X + 4 multiplied by the derivative of Y cubed plus the derivative of Y, and we're going to set it equal to the derivative of 9. The derivative of X based on the power rule is 1, so we get -2. Plus 4 multiplied by 3 squad. This is the derivative of Y cubed based on the power rule, and Y is a function of X, so we also multiplied by Y and then we add Y. We set it equal to 0 because the derivative of a constant is 0. And now we factor out, we get 4 multiplied by 3 Y2 which is 12 Y quad. We add 1. And then we have negative 2, which we can move to the right, and this gives us equal. 2 So essentially now we want to solve for Y. This is the general expression of the slope of the tangent line. So we take 2 and divide that by 12 Y2 + 1. Let's recall that a horizontal tension line satisfies Y equals 0, and now this is a rational function. So whenever it is equal to 0, we want to set our numerator equal to 0 and our denominator not equal to 0. So now let's notice that our numerator doesn't involve any x or y variables. So 2 is never equal to 0 because it's independent of X or Y. It is simply a constant, and because it is never equal to 0, we can conclude that the correct answer to this problem corresponds to the answer to D, none because the derivative will never be zero. Thank you for watching.

Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.

Key Concepts

Tangent Lines