How to graph f ' and justify?

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1 Answer
Shwetank Mauria
Jan 31, 2018

Please see below.

Explanation:

Let us first observe tha function. A few observations are:

  1. It is always positive.
  2. Its value is #0# at #x=+-a# i.e. at #(-a,0)# and #(a,0)#

Hence, it is of the type #f(x)=g(x)(x+a)(x-a)#. We have taken it as #g(x)#, which is not a constant but a function of #x# as otherwise it would have been a parabola.

At #x=+-a#, we have a minima too, that means #f'(x)=0# at #(-a,0)# and #(a,0)#, which means #(x-a)# and #(x+a)# are factors of #f'(x)# too. Can we have it as #f(x)=k(x+a)^2(x-a)^2=k(x^2-a^2)^2# i.e. #k(x^4-2a^2x^2+a^4)#? It is evident that it is positive when #x=0#. Further, as #f(x)=k(x^2-a^2)^2#, #f’(x)=4kx(x^2-a^2)# i.e. #f’(x)=0# at #(-a,0)# and #(a,0)#.

Before we conclude, let us observe the slope of #f(x)# too as it tells us more about #f'(x)#. Observe that as we move #x# from LHS, increasing #x# slowly across #x#-axis:

  1. Initially slope is steeply negative but continues to increase and at #(-a,0)# it reaches #0#. It continues to increase further till somewhere between #(-a,0)# and #(0,0)#, where we have a point of inflection, where #f''(x)=0#. In other words, #f’’(x)>0#, till point of inflection and while #f’(x)# reaches a local maxima, #f’’(x)=0# at this point between #(-a,0)# and #(0,0)#.

  2. Beyond this point of inflection, where #f’’(x))=0#, slope i.e. #f'(x)# falls, till it reaches #0# again at #(0,0)#. It continues to fall, till next point of inflection is reached between #(0,0)# and #(a,0)#. At this point of inflection, #f’(x)# reaches a local minima where #f’(x)<0# and #f’’(x)=0#.

  3. As we move further right beyond this point of inflection, #f’(x)# continues to increase reaching #0# at #(a,0)# and beyond this #f’(x)# continues to increase.

As #f’(x)=4kx(x^2-a^2)=4kx^3-4ka^2x# and #f’’(x)=12kx^2-4ka^2=4k(3x^2-a^2)#, these trends are indeed as seen in the graph under consideration,

and hence if #f(x)# is (here #k=1/4# and #a=sqrt3#)

graph{1/4(x^4-6x^2+9) [-5, 5, -1.5, 3.5]}

#f’(x)# appears as shown below.

graph{x(x^2-3) [-10, 10, -5, 5]}

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