Graphing rational functions ?Use the guidelines of this section to make a complete graph of f.

Solution: Step1 Given function is f(x)= x +12 2x+1 The zero of the denominator is x=-0.5 so, the domain is { x; x/ -0.5} This function consists of an even function divided by an odd function. The product of even and odd function is odd. Therefore, the graph is symmetric about the origin. Step2 Differentiate the given equation to find f’(x) we get, f’(x)= (2x)/(2x+1) - (2(x^2+12))/(2x+1)^2 = (2x(2x+1)-2(x^2+12))/(2x+1)^2 = (4x^2+2x-2x^2-24)/(2x+1)^2 = (2x^2+2x-24)/(2x+1)^2 = (2(x^2+x-12))/(2x+1)^2 Again differentiate f’(x) to find f’’(x) we get, f’’(x)=(4x+2)/(2x+1)^2 - (4(2x^2+2x-24))/(2x+1)^3 = ((4x+2)(2x+1)-8x^2-8x+96)/(2x+1)^3 = (8x^2+4x+4x+2-8x^2-8x+96)/(2x+1)^3 = 98/(2x+1)^3 Step3 To get extreme values we have to use f’(x)=0 => (2(x^2+x-12))/(2x+1)^2 =0 =>2(x^2+x-12)=0 => x^2+x-12=0 =>x^2+4x-3x-12=0 => x(x+4)-3(x+4)=0 =>(x-3)(x+4)=0 =>x=3,-4 Critical points...