Question

Find the critical numbers of the function anddescribe the behavior of f at these numbers. (List youranswers in increasing order.)

f(x) =x¹⁰(x -1)⁹

At the function has ( local min orlocal max or neither )
At the function has ( local min orlocal max or neither )
At the function has ( local min orlocal max or neither )

Answer 1

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Set f '(x) =0 and solve for x

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which are critical numbers
At x=0 ==> f '(x) change sign from from positive to negative==> f(x) has local maximum at x=0
At x=10/19 ==> f '(x) changes sign from negative to positive==> f(x) has local minimum at x=10/19
At x=1 ==> f '(x) does not change sign ==> f(x) has no localmaximum nor local minimum

Thus,

Critical numbers: x=0;x=10/19;x=1

At x=0 the function has localmax
At x=10/19 the functionhas local min
At x=1 the function has neither local maxnor local min

Answer 2

Set it equal to zero:

you can factor out an x^9 and an (x-1)^8 which means you havex=0 and x=1 as critical points so far:

___________

Now test points below, between, and above the criticalpoints:

We'll use x=-1, 1/2, 3/4, and 2

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this means you have a max at x=0, a min at x=10/19, andneither at x=1