`f(x) =x^4 - 2x^2 + 3`

(a) To solve, take the derivative of the given function.

`f'(x) =4x^3 - 4x`

Then, set the derivative equal to zero.

`0=4x^3-4x`

Factor the right side of the equation.

`0=4x(x^2-1)`

`0=4x(x-1)(x+1)`

Set each factor equal to zero. And isolate the x in each equation.

Factor 1:

`4x=0`

`x=0`

Factor 2:

`x-1=0`

`x=1`

Factor 3:

`x + 1 =0`

`x=-1`

So, f(x) has three critical numbers. These are x=-1, x=0 and x=1. The intervals formed by these three critical numbers are:

`(-oo, -1)` `(-1,0)` `(0,1)` and `(1,oo)` .

To determine which among the intervals is the function increasing or decreasing, assign a test value for each. Plug-in the test value to the derivative

`f'(x) = 4x^3-4x.`

If the result of f'(x) is negative, the function is decreasing in that interval. And if the result is positive, the function is increasing in that interval.

For the first interval (-inf,-1), let the test value be x=-2.

`f'(-2) = 4(-2)^3 - 4(-2)=-24 ` (Decreasing)

For the second interval...

(The entire section contains 650 words.)

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