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.
A local minimum is a point on a graph where the function has the lowest value within a small interval around that point. It can be visualized as the bottom of a valley on a curve.
To find the local minimum of a function, you can take the derivative of the function and set it equal to zero. Then, solve for the value of x that makes the derivative equal to zero. This value of x will be the x-coordinate of the local minimum. You can also graph the function and visually determine the location of the local minimum.
The given function is a fourth-degree polynomial function. It can also be written as x^4-9x^3+9x^2+5x-4 = x^4-9x^3+9x^2+5x+(-4), where the coefficients of x^4, x^3, x^2, x, and the constant term are -1, -9, 9, 5, and -4 respectively.
To confirm the location of the local minimum, you can plug in the x-coordinate of the local minimum into the original function and see if the resulting y-value is the lowest within a small interval around that point. You can also check your work by taking the second derivative of the function, which should be positive at the local minimum point.
Yes, a function can have multiple local minima if it has multiple valleys or dips in the graph. These local minima can have different y-values, but they all have the lowest y-values within their respective small intervals.