Analysis of Special Points on the Graph of a Function

Example 1.  

[Maple Plot]

f(x) = 3*x^4-4*x^3-12*x^2+8

f(x) = 0

3*x^4-4*x^3-12*x^2+8 = 0

`x-intercepts on graph of f: `

.7756334518, 2.685233572

[Maple Plot]

`f '`(x) = 12*x^3-12*x^2-24*x

`f '`(x) = 0

12*x*(x+1)*(x-2) = 0

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

-1, 0, 2

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

`f '`(-1.2) = -9.216, `f '`(-1) = 0, `f '`(-.8) = 5.376

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

`f ' sign pattern is:  -`, 0, `+`

`bump 2 on graph of f: `*[0, f(0)] = [0, 8]

`f '`(-.2) = 4.224, `f '`(0) = 0, `f '`(.2) = -5.184

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

`f ' sign pattern is:  +`, 0, `-`

`bump 3 on graph of f: `*[2, f(2)] = [2, -24]

`f '`(1.8) = -12.096, `f '`(2) = 0, `f '`(2.2) = 16.896

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

`f ' sign pattern is:  -`, 0, `+`

[Maple Plot]

`f ''`(x) = 36*x^2-24*x-24

`f ''`(x) = 0

36*x^2-24*x-24 = 0

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

-.5485837704, 1.215250437

`pip 1 on graph of f: `*[-.5485837704, f(-.5485837704)] = [-.5485837704, 5.320744626]

`f ''`(-.6) = 3.36, `f ''`(-.5485837704) = 0., `f ''`(-.5) = -3.00

`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: `*[1.215250437, f(1.215250437)] = [1.215250437, -10.35778167]

`f ''`(1.0) = -12.00, `f ''`(1.215250437) = .1e-7, `f ''`(1.4) = 12.96

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

`f '' sign pattern is:  -`, 0, `+`