The discussion centers on finding the local minima of the polynomial function x^4 - 9x^3 + 9x^2 + 5x - 4. The identified local minima are at x = -0.21 and x = 5.96. While 5.96 is confirmed as the absolute minimum due to yielding the lowest y-value on the open interval, the conversation clarifies that all absolute minima are also local minima, but not all local minima are absolute. This distinction is crucial for understanding the behavior of polynomial functions.
PREREQUISITESStudents studying calculus, mathematicians analyzing polynomial functions, and educators teaching concepts of minima and maxima in mathematics.
ideasrule said:An absolute minimum is a local minimum; if it's a minimum across the entire interval, it's obviously a minimum locally.
Dick said:You aren't wrong about x=5.96 being the absolute minimum. But the problem is asking for local minima. There's more than one.
koudai8 said:I just thought the other value is the absolute minimum, not the local minimum, so I didn't include it as the local minimum. So, the absolute is ALWAYS also a local one?
Dick said:Sure it is. ideasrule has it correct.