a cubic polynomial function f is defined by f(x)=4x^3 +ax^2 +bx +k where a, b, and k are constants. the function f has a local min at x=-1 and the graph of f has a point of inflection at x=-2. find the values of a and b. if the integral from [0, 1] f(x)dx=32, what is the value of k? please help and show all steps!

Question

turingtest · Answer

the first derivative is$$f'(x)=12x^2+2ax+b$$a minimum here means that$$f'(-1)=12-2a+b=0$$the second derivative is$$f''(x)=24x+2a$$an inflection point here means that$$f''(-2)=-48+2a=0$$use this expression to find a, then use that value of a to find b from f'(-1)
then simply do the integral and solve for k