- Jun 26, 2006

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f(x) = x^4 - 4x^2 + k

For what value of k does the function have two real zeroes, each with a multiplicity of two?

So from there I figured out that since the multiplicity was 2, the function would just touch the x-axis at the zeroes, but not cross. So I took the derivative of the function and set y to 0 to find the extrema, which were +- sqrt(2). Then I plugged them back into the original equation, and set y to 0 to get k. I believe this is the correct solution, but how would you do this without derivatives? Basically, how do you get the extrema of an equation that has a degree greater than 2 without finding where the derivative of the function = 0?