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