d²f/dx² = 36x^2 - 216x - 3024a) Roots → f(x) = 0... 3x^2 ( x^4 - 12x^3 - 504 ) = 0 ← use the quadratic formulaor x = { 0, 6 ± 6 √15 }b) Y- intercept = f(0)........
f '(-1) = 12(-1)^3 - 108(-1)² - 3024(-1) = 2904
f '(1) = -3120
f '(22) = 8976
So we see that the sign changes from negative to positive at -12 and 21, and the sign changes from positive to negative at 0. Thus, there is a minimum at x=-12 and x=21, and a maximum at x=0. (As in part c, plug those x-coordinates back into f(x) to find the y-coordinates)
Hope that helps :)
... f(x) = 3x^4 - 36x^3 - 1512x^2
or f(x) = 3x^2 ( x^4 - 12x^3 - 504 )
... df/dx = 12x^3 - 108x^2 - 3024x
or df/dx = 12x (x + 12) (x-21)
... d²f/dx² = 36x^2 - 216x - 3024
a) Roots → f(x) = 0
... 3x^2 ( x^4 - 12x^3 - 504 ) = 0 ← use the quadratic formula
or x = { 0, 6 ± 6 √15 }
b) Y- intercept = f(0)
... 3(0)^2 ( (0)^4 - 12(0)^3 - 504 ) = 0
c) Points of Inflection → d²f/dx² = 0
... 36x^2 - 216x - 3024 = 0 ← use the quadratic formula
or x = { 3 ± √ 93 }
d & e)
... df/dx = 12x (x + 12) (x - 24) = 0 at max/min
or x = { -12, 0, 24 }
d²f/dx² [ at x = -12 ] = 36(-12)^2 - 216(-12) - 3024 = 4752 > 0 ← relative minimum
d²f/dx² [ at x = 0 ] = 36(0)^2 - 216(0) - 3024 = 0 ← relative maximum
d²f/dx² [ at x = 24 ] = 36(24)^2 - 216(24) - 3024 = 12528 > 0 ← relative mimimum
See Graph: http://www.wolframalpha.com/input/?i=f%2…
