If they exist, how do you find them & what is the rel min/max of f(x)=(4x^2-12x)/(x^2-2x-3)? How do you find the point of inflection, if there is one? I am doing a function summarty of this particular rational function and when setting f'(x) is set equal to 0, it becomes 4=0, which is impossible; the same problem occurs when setting f"(x) is also set equal to 0. The shape of the graph make me think there is no rel min or max nor a point of inflection, but is it concave up/down and if so, where? Con up (-infinit, -1) and Con down (-1, infinit)?

Expert Answers

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You may simplify the function to its lowest terms, hence you should write the factored forms of numerator and denominator such that:

f(x) = 4x(x-3)/(x+1)(x-3) = 4x/(x+1)

You need to remember that solving the equation f'(x)=0 you may find information about minimum/maximum of function f(x) such that:

f'(x) = (4(x+1)-4x(x+1)')/((x+1)^2)

f'(x) = (4x+4-4x)/((x+1)^2)

f'(x) = 4/((x+1)^2)

Since f'(x) != 0 for any value of x, then the function has no such points as maximum,minimum or inflection points.

The line x=-1 becomes the vertical asymptote to graph of the function as is presented in the sketch below:

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