determine the constants a, b, and c in order for the function f(x)= x^3 +ax^2 +bx+c to have a relative maximum at x=1 and a point of inflection at x=2. find a relative minimum of the function given the found values of a and b, provided that f(0)=6. please help and show all steps!

Question

nikvist · Answer

$$f(x)=x^3+ax^2+bx+c\quad,\quad f(0)=6=c$$$$f'(x)=3x^2+2ax+b\quad,\quad f'(1)=3+2a+b$$$$f''(x)=6x+2a\quad,\quad f''(2)=12+2a=0\quad\Rightarrow\quad a=-6$$$$f'(1)=3-12+b=b-9=0\quad\Rightarrow\quad b=9$$$$f'(x)=3x^2-12x+9=3(x-1)(x-3)$$$$f_{\min}=f(3)=3^3-6\cdot 3^2+9\cdot 3+6=27-54+27+6=6$$ http://www.wolframalpha.com/input/?i=y%3Dx^3-6x^2%2B9x%2B6

anonymous · Answer

nikvist's solution is correct. A two page PDF problem expression solution is attached.