Graphing polynomials ?Sketch a graph of the following polynomials. identify local extrema, inflection points?, and x- ?and y-intercepts when they exist. ? ? ?f(?x) =?? 4 + 4? 3 ? 12?x2

Solution: Step1 Given that f(x)=3*x^4+4*x^3-12*x^2 Differentiate the given equation to find f’(x) we get, f’(x)= 3*4*x^3+4*3*x^2-12*2*x = 12*x^3+12*x^2-24x Again differentiate f’(x) to find f’’(x) we get, f’’(x)=12*3*x^2+12*2*x-24 = 36*x^2+24*x-24 Step2 To get extreme values we have to use f’(x)=0 => 12*x^3+12*x^2-24x=0 =>12x(x^2+x-2)=0 => x(x^2+2x-x-2)=0 => x(x-1)(x+2)=0 Critical points are 0,1 and -2 Step3 To find inflection points we have to use f’’(x)=0 =>36*x^2+24*x-24=0 =>6(6*x^2+4x-4)=0 =>6*x^2+4x-4=0 =>2 (3*x^2+2x-2)=0 =>3*x^2+2x-2=0 => x=(-2± 4 + 24)/6 =>x=-1.21,0.55 Evaluate f’’(x) at the critical points we get, f”(0 )=36*x^2+24*x-24 = 36*(0)^2+24*0-24 =-24<0 has a local maximum at x=0 And f”(1)= 36*x^2+24*x-24 = 36*(1)^2+24*1-24 = 36-0=36>=0...