Analysis of Special Points on the Graph of a Function

Example 3.  

[Maple Plot]

f(x) = x^6+x^5-7*x^4

f(x) = 0

x^4*(x^2+x-7) = 0

`x-intercepts on graph of f: `

-3.192582404, 0., 2.192582404



[Maple Plot]

`f '`(x) = 6*x^5+5*x^4-28*x^3

`f '`(x) = 0

x^3*(6*x^2+5*x-28) = 0

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

-2.616729797, 0., 1.783396464

`bump 1 on graph of f: `*[-2.616729797, f(-2.616729797)] = [-2.616729797, -129.8463741]

`f '`(-2.8) = -110.63808, `f '`(-2.616729797) = 0., `f '`(-2.4) = 75.20256

`bump 1 is a local min (valley), since`

`f ' sign pattern is:  -`, 0, `+`

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

`f '`(-.2) = .23008, `f '`(0) = 0, `f '`(.2) = -.21408

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

`f ' sign pattern is:  +`, 0, `-`

`bump 3 on graph of f: `*[1.783396464, f(1.783396464)] = [1.783396464, -20.59637707]

`f '`(1.5) = -23.62500, `f '`(1.783396464) = 0., `f '`(1.9) = 21.67444

`bump 3 is a local min (valley), since`

`f ' sign pattern is:  -`, 0, `+`



[Maple Plot]

`f ''`(x) = 30*x^4+20*x^3-84*x^2

`f ''`(x) = 0

2*x^2*(15*x^2+10*x-42) = 0

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

-2.039531186, 0., 1.372864519

`pip 1 on graph of f: `*[-2.039531186, f(-2.039531186)] = [-2.039531186, -84.43593474]

`f ''`(-2.2) = 83.2480, `f ''`(-2.039531186) = 0., `f ''`(-1.8) = -73.8720

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

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

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

`f ''`(-.2) = -3.4720, `f ''`(0) = 0, `f ''`(.2) = -3.1520

`pip 2 is not an inflection point because the slope function f ' does not`

`have a local max or a local min at pip 2, since f '' sign pattern is:  -`, 0, `-`

`pip 3 on graph of f: `*[1.372864519, f(1.372864519)] = [1.372864519, -13.29408443]

`f ''`(1.1) = -31.0970, `f ''`(1.372864519) = 0., `f ''`(1.5) = 30.3750

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

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