How do you sketch the graph that satisfies f'(x)>0 when -1<x<3, f'(x)<0 when x<-1 and when x>3, and f(-1)=0, f(3)=4?

1 Answer
Oct 13, 2017

See sample drawing in explanation below. Example curve:
#f(x) = -x^3+3x^2+9x+2#

Explanation:

Sketching the graph is fairly straightforward once you interpret what each piece of the problem tells you about the shape of the graph.

#f'(x)>0# when # -1 < x < 3 #

The first derivative tells you whether the graph is increasing ("going upwards") or decreasing ("going downwards"). If #f'(x)>0# (positive), then the graph is increasing. This statement tells us that from -1 to 3, the function will be increasing in y-values.

#f'(x)<0# when #x < -1# and #x > 3#

This statement says that #f'(x)<0# (negative) both to the left of -1 and to the right of 3. Visually, this means that the left edge of the number line to the left of -1 is going downward in y-values. On the right edge of the number line after 3, it again is falling in y-values.

#f(-1)=0# and #f(3)=4#

This gives us not just two points to draw that the graph goes through, but specifically the two points where the increasing/decreasing intervals change at.

On paper, on the left edge of the paper, draw the graph falling downwards from the upper-left corner of the paper until you hit the point #(-1,0)#, then curve it back upwards until you hit the point #(3,4)#, after which point make it fall downwards towards the lower-right corner of the paper.

Note: We aren't told anything about the second derivative, so we cannot be sure if the graph sections are concave up or down. We also don't know for sure what #f'(-1)# and #f'(3)# is (if even defined), so we don't know if the graph smoothly curves through those points or bounces sharply off.

Here's a possible example of what the graph might look like. For additional reading, look below the graph here for an example of how I derived this function for graphing.

graph{-x^3+3x^2+9x+2 [-6, 6, -16, 35]}

Example Function

Let's assume the function is a polynomial for convenience. Since we know that the left edge/tail of the graph points upward, and the right edge/tail of the graph points downward, the degree of the polynomial must be odd. ("End behavior of polynomials")

Further, there are two known critical points for the derivative (-1 and 3), which just so happen to be a minimum and maximum respectively. Thus, we know that #f'(-1)# and #f'(3)# are either 0 or undefined at those points. We can assume there are no further critical points for argument sake. We can also assume that the derivative is 0 at each for convenience sake.

This implies that -1 and 3 are zeroes of #f'(x)#, or put another way , that #(x+1)# and #(x-3)# are factors of #f'(x)#. Thus:

#f'(x) = A(x+1)(x-3)#, where #A# is some non-zero constant.

So:

#f'(x) = A(x^2-2x-3) = Ax^2-2Ax-3A#

However, we're told that #f'(x)<0# for #x<-1#. Let's examine #x = -2#:

#f'(-2) = A(-2)^2-2A(-2)-3A = 4A+4A-3A#

# = 5A < 0 => A < 0#

We know that #A < 0#. Continuing:

#f(x) = int f'(x) dx = int (Ax^2-2Ax-3A) dx#

# = A/3x^3 -Ax^2 -3Ax + C# where #C# is some constant.

Any choice of negative #A# and any #C# will define a graph that matches the criteria provided in this problem. I chose #A=-3# and #C = 2# to get #f(x) = -x^3+3x^2+9x+2# for my graph above.