Graphing polynomials ?Sketch a graph of the following polynomials. identify local extrema, inflection points, and x?- ?and y-intercepts when they exist.
Solution: Step1 Given that f(x)=1/15 x^3-x+1 Differentiate the given equation to find f’(x) we get, f’(x)= 1/15*3*x^2 - 1 +0 = x^ -1 Again differentiate f’(x) to find f’’(x) we get, f’’(x)=2x/5-0=2x/5 Step2 To get extreme values we have to use f’(x)=0 => 3/15 x^2-1=0 =>3/15 x^2=1 => x^2= 1*15/3=5 => x=± 5 Critical points are - 5 and + 5 Step3 To find inflection points we have to use f’’(x)=0 =>*x=0 =>x=0 Evaluate f’’(x) at the critical points we get, f”(- 5 )= * x= ( 5 ) =(-2*2.24)/5= -4.48/5 = -0.896<0 f(x) has a local maximum at x=- 5 And f”( 5 )= *x = ( 5 ) = (2*2.24)/5=4.48/5=0.896 Has a local minimum at x= 5 Step4 The corresponding function values are f(- 5 )= 1/15x^3-x+1 = 1/15(- 5 )^3-(- 5 )+1 =-5 5 /15+ 5 +1 = -2.24/3+2.24+1 =-0.75+2.24+1 =3.34-0.75=2.59 And f( 5 )= 1/15x^3-x+1 = 1/15( 5 )^3-( 5 )+1 =5 5 /15- 5 +1 = 2.24/3-2.24+1 =0.75-2.24+1 = 1.75-2.24=-0.49 Finally we see that f”(x) changes sign at x=0 f(0)=1 Therefore the inflection point is (0,1) And the local maximum at (- 5,2.59) , Local minimum at ( 5,-0.49) Step5
