Can a parabola have more than one tangent at a point?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Consider a curve given by y = f(x).

The derivative of the function at any point x = a is given by

f'(a) = `lim_(h->0)(f(a+h) - f(a))/h`

Differentiation is also equivalent to taking two points corresponding to f(a) and f(a+h) and drawing a line between them. [f(a+h)-f(a)]/h is equal to the slope of the line between f(a) and f(a +h) and is the slope of the tangent at x = a as h tends to 0.

For any function, f'(a) either exists if the function is continuous or it does not exist where there are points of discontinuities.

A parabola is a continuous function, with the derivative existing at all points and having a unique value.

Therefore a parabola cannot have more than tangent at any point.