The absolute minimum and maximum point of a function is the point where the function has the least and highest value, respectively.
- The absolute minimum is 0
- The absolute maximum is 42875
The function is given as:
[tex]\mathbf{f(x) = (x^2 - 1)^3.\ [-1,6]}[/tex]
There are several ways to do this;
One of them is by substituting the x-values in the function.
So, we have:
[tex]\mathbf{f(-1) = ((-1)^2 - 1)^3 = 0}[/tex]
[tex]\mathbf{f(0) = (0^2 - 1)^3 = -1}[/tex]
[tex]\mathbf{f(1) = (1^2 - 1)^3 = 0}[/tex]
[tex]\mathbf{f(2) = (2^2 - 1)^3 = 27}[/tex]
[tex]\mathbf{f(3) = (3^2 - 1)^3 = 512}[/tex]
[tex]\mathbf{f(4) = (4^2 - 1)^3 = 3375}[/tex]
[tex]\mathbf{f(5) = (5^2 - 1)^3 = 13824}[/tex]
[tex]\mathbf{f(6) = (6^2 - 1)^3 = 42875}[/tex]
By comparing the above values, we have the minimum and maximum are:
[tex]\mathbf{f(0) = -1}[/tex]
[tex]\mathbf{f(6) = 42875}[/tex]
Hence, the absolute minimum and maximum are 0 and 42875, respectively.
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