# `f(x) = x^4 - 2x^2 + 3` (a) Find the intervals on which `f` is increasing or decreasing. (b) Find the local maximum and minimum values of `f`. (c) Find the intervals of concavity and the inflection points.

## Expert Answers `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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