What would the f' and f" graphs look like?

Shown below if a graph of function f, a fourth degree polynomial function. On
the axes shown sketch, approximately, the graphs of f'and f" .
enter image source here

1 Answer
Douglas K.
Feb 7, 2018

The x coordinate of the point in the center is clearly 0.
Let the x coordinate of the point on the left be #x_1#.
Let the x coordinate of the point on the right be #x_2#.

These x coordinates are the zeros of #f'(x)#:

Writing #f'(x)# in factored form:

#f'(x) = x(x-x_1)(x-x_2)#

Multiply the factors:

#f'(x) = x(x^2-(x_1+x_2)x+x_1x_2)#

#f'(x) = x^3-(x_1+x_2)x^2+x_1x_2x#

#f(x)# is the integral of #f'(x)#:

#f(x) = 1/4x^4-1/3(x_1+x_2)x^3+1/2x_1x_2x^2+c#

#f''(x)# is the derivative of #f'(x)#

#f''(x) = 3x^2-2(x_1+x_2)x+x_1x_2#

To give you an idea of the shapes of the two derivatives, I have graphed, #f(x)#, #color(red)(f'(x))# and #color(blue)(f''(x))#, using the https://www.desmos.com/calculator online graphic calculator:

www.desmos.com/calculator

In an attempt to make #f(x)# look like the sketch, I have chosen, #x_1=-1.5#, #x_2=1#, and #c=2#; the Desmos calculator uses these same parameters in the graphs for #color(red)(f'(x))# and #color(blue)(f''(x))#.

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