Given: f(x) = y = 3x^4 - 4x^3 = x^3(3x - 4)
Find the first derivative and set y' = 0 to find the critical points:
" "y' = 12x^3 - 12x^2 = 0
" "y' = 12x^2(x - 1) = 0
" "critical values: " "x = 0, x = 1
" "f(0) = 0; " " f(1) = 3(1)^4 - 4(1)^3 = -1
" "critical points: (0, 0), (1, -1)
Find the second derivative and evaluate it at the critical values to find relative minimums or relative maximums:
y'' = 36x^2-24x
y''(0) = 0 " "=> need to use the first derivative test to find out if this point is a relative min., max. or inflection.
y''(1) = 36(1)^2 - 24(1) = 12 > 0 => relative min. at (1, -1)
Find inflection points by setting y'' = 0:
y'' = 36x^2-24x =0
y'' = 12x(3x - 2) = 0 => inflections at x = 0, x = 2/3
f(2/3) = 3(2/3)^4 - 4(2/3)^3 = 3*16/81 - 4*8/27 = -16/27
inflection points (0, 0), (2/3, -16/27)
To graph:
Fourth order functions have end conditions both positive. Put a point at (0,0) and a point at the relative minimum (1, -1)
Start at the upper left and create a horizontal inflection at (0.0), then turn downward to the relative minimum, then curve upward.
graph{3x^4-4x^3 [-5, 5, -5, 5]}
