Curve Sketching

Turning Points


Given the function f(x)=x28x+12x2+6x16 f(x) = \frac{ x^2-8x + 12}{x^2+6x - 16} , what is limx2f(x)\displaystyle{\lim_{ x \rightarrow 2 } f(x)} ?

How many integers kk are there such that the function f(x)=x3+kx2+3x+2f(x)=x^3+kx^2+3x+2 has no turning points?

Let f(x)=x36x2+14x+9.f(x)=x^3-6x^2+14x+9. What is the sum of the xx-coordinates of turning points such that f(x)f(x) switches from a decreasing function to an increasing function?

A polynomial of degree 2525 has mm real roots and nn turning points. What is the maximum value of m+nm+n?

What is the sum of all the xx-coordinates of the turning points in the graph of f(x)=2x3+18x230x+9?f(x)=-2x^3+18x^2-30x+9?


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