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 11 X plus Y to the power of 4 minus 2y equals 3? Justify your answer. We're given 4 answer choices. A says yes, since the first derivative is equal to 0 at X equals 2 and X equals 6. There are two horizontal tangent lines. B says yes, since the first derivative is equal to 0 at X equals 2 and 3. There are two horizontal tension lines. Yes, since the first derivative is equal to 0 at X equals 2, there is one horizontal tangent line, and D says none because the first derivative will never be zero. So essentially we have to find the first derivative, right? We are going to differentiate this expression implicitly we're given 11 X plus Y to the power of 4th minus 2y. We're going to set this derivative equal to the derivative of 3. Now, for the left hand side, we can essentially say that we're differentiating 11 X. Plus the derivative of y to the power of 4th minus 2 multiplied by the derivative of Y, and this is going to be equal to the derivative of 3. So the derivative of X is 1, we get 11 plus. The derivative of y is the power of 4th based on the power rule is 4 y is the power of 3, and because y is a function of X, we multiply by y based on the chain rule we subtract 2 Y and this must be equal to 0 because the derivative of a constant is 0. And now we factor out y. We got 4 y cubed minus 2 + 11, so we can essentially move 11 to the right and we're going to get equals -11. And this allows us to solve for Y, which is equal to -11 divided by or Y cubed minus 2. And if we want to solve for horizontal tangent lines, we have to recall that the first derivative must be equal to 0. So we set this expression to 0. And notice that this is a rational expression. So essentially, we want to make sure that our numerator is equal to 0 and our denominator is not equal to 0. So, considering the numerator, which is the most important part, we understand that it has -11. And -11 is never equal to 0. It's independent of X or Y. It is basically constant, and this means that it doesn't matter what X or Y is, our numerator will never be equal to 0, and the correct answer choice corresponds to D. There are no horizontal tangent lines because D Y divided by D X, which is the first 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