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Show that the curve
has no tangent line with slope 4.Solution:
Consider the given curve
As we know that the slope of any curve is equal to the first derivative of the equation of any curve with respect to the independent variable which is x in most cases. Take the derivative of the given equation with respect to x, we have
If the slope of a curve is dy/dx = 4 which is also the slope of a tangent line, then the above equation becomes
From the resulting equation, the values of x will be the x values of the intersection of a curve and a tangent line. By using quadratic formula, the values of x are
Since the values of x are imaginary numbers or complex numbers, then there's no tangent line for the given curve. The given curve did not intersect with the tangent line of a given slope.