Find a cubic function, that has a local maximum at the point (4,200), an inflection point at x-7 and f^(3) (-5)=6. The function given is f(x)=ax^3+bx^2+cx+d. f(x)=?
f(x) = ax^3 + bx^2 + cx + d f '(x) = 3ax^2 + 2bx + c f ''(x) = 6ax + 2b f '''(x) = 6a f '''(-5) = 6 6 = 6a a = 1 f ''(x) = 6x + 2b f ''(7) = 0 0 = 6(7) + 2b b = -21 f '(x) = 3x^2 - 42x + c f '(4) = 0 0 = 3(4^2) - 42(4) + c 0 = 48 - 168 + c 0 = -120 + c c = 120 f(x) = x^3 - 21x^2 + 120x + d f(4) = 200 200 = 4^3 - 21(4^2) + 120(4) + d 200 = 64 - 336 + 480 + d 200 = 208 + d d = 12 f(x) = x^3 - 21x^2 + 120x + 12