Does any tangent to the curve y=�ãx cross the x-axis at x= -1? If so, find an equation for the line and the point of tangency. If not, why not?

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To determine if any tangent to the curve y = x^2 crosses the x-axis at x = -1, we will need to find the derivative of the equation and check if the derivative evaluated at x = -1 is zero.

First, we take the derivative of y = x^2 with respect to x. Since the power rule states that the derivative of x^n is n*x^(n-1), the derivative of x^2 is:

dy/dx = 2x

Next, we evaluate the derivative at x = -1:

dy/dx = 2(-1)
= -2

Since the derivative is not equal to zero (dy/dx ≠ 0), it means that the tangent line to the curve y = x^2 does not cross the x-axis at x = -1.

If the derivative had been equal to zero (dy/dx = 0) at x = -1, it would indicate that there is a point of tangency on the curve at that x-coordinate. In that case, we could find the equation of the tangent line using the point-slope form, since we know the slope of the tangent line at that point.

However, since the derivative is not zero, no tangent line intersects the x-axis at x = -1 for the curve y = x^2.