Analysis of Special Points on the Graph of a Function

Example 2.  

[Maple Plot]

f(x) = -x^6+3*x^4-3*x^2+1

f(x) = 0

-(x-1)^3*(x+1)^3 = 0

`x-intercepts on graph of f: `

-1.000000000, 1.


[Maple Plot]

`f '`(x) = -6*x^5+12*x^3-6*x

`f '`(x) = 0

-6*x*(x-1)^2*(x+1)^2 = 0

`x-coord. of bumps on graph of f: `

-1.000000000, 0., 1.

`bump 1 on graph of f: `*[-1, f(-1)] = [-1, 0]

`f '`(-1.2) = 1.39392, `f '`(-1) = 0, `f '`(-.8) = .62208

`bump 1 is a plateau, since`

`f ' sign pattern is:  +`, 0, `+`

`bump 2 on graph of f: `*[0, f(0)] = [0, 1]

`f '`(-.2) = 1.10592, `f '`(0) = 0, `f '`(.2) = -1.10592

`bump 2 is a local max (peak), since`

`f ' sign pattern is:  +`, 0, `-`

`bump 3 on graph of f: `*[1, f(1)] = [1, 0]

`f '`(.8) = -.62208, `f '`(1) = 0, `f '`(1.2) = -1.39392

`bump 3 is a plateau, since`

`f ' sign pattern is:  -`, 0, `-`


[Maple Plot]

`f ''`(x) = -30*x^4+36*x^2-6

`f ''`(x) = 0

-6*(x-1)*(x+1)*(5*x^2-1) = 0

`x-coord. of pips (possible inflection points) on graph of f: `

-1., -.4472135955, .4472135955, 1.

`pip 1 on graph of f: `*[-1, f(-1)] = [-1, 0]

`f ''`(-1.2) = -16.3680, `f ''`(-1) = 0, `f ''`(-.8) = 4.7520

`pip 1 is an inflection point which corresponds to a local min of the slope function f ',`

`since f '' sign pattern is:  -`, 0, `+`

`pip 2 on graph of f: `*[-.4472135955, f(-.4472135955)] = [-.4472135955, .5120000000]

`f ''`(-.6472135955) = 3.815925471, `f ''`(-.4472135955) = 0., `f ''`(-.2472135955) = -3.911925465

`pip 2 is an inflection point which corresponds to a local max of the slope function f ',`

`since f '' sign pattern is:  +`, 0, `-`

`pip 3 on graph of f: `*[.4472135955, f(.4472135955)] = [.4472135955, .5120000000]

`f ''`(.2472135955) = -3.911925465, `f ''`(.4472135955) = 0., `f ''`(.6472135955) = 3.815925471

`pip 3 is an inflection point which corresponds to a local min of the slope function f ',`

`since f '' sign pattern is:  -`, 0, `+`

`pip 4 on graph of f: `*[1, f(1)] = [1, 0]

`f ''`(.8) = 4.7520, `f ''`(1) = 0, `f ''`(1.2) = -16.3680

`pip 4 is an inflection point which corresponds to a local max of the slope function f ',`

`since f '' sign pattern is:  +`, 0, `-`