Question

How many tangents does the graph of f(x)=(x-1)^2(x-3)^2 have?

I found out the derivative of f(x)=(x-1)^2(x-3)^2

f'(x)=4x^3 - 24x^2 + 44x -24

What do I do next?

Answer 1

An infinite number of points lie on the curve and there is a tangent line at every point. So may they were asking how many times the tangent not exists. The tangent would not exist if the derivative was undefined at some point. Since the derivative is a polynomial it is defined everywhere. So no matter how you look at it there are infinitely many tangent lines. Since the derivative is a third order polynomial the slope of the tangent will be 0 3 times.