SOLUTION: sketch a graph of a function f(x) on the interval -2 (less than or equal to) x (less than or equal to) 2 with the following properties. Label any points on the x-axis or y-axis whi

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Question 1138898: sketch a graph of a function f(x) on the interval -2 (less than or equal to) x (less than or equal to) 2 with the following properties. Label any points on the x-axis or y-axis which are important.
f is odd.
f"(x) is positive for all x>0.
f'(0) is negative.
f'(1) = 0.
f(2) = 0.
thanks for helping!!
Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website!

(1) f(x) is odd.

That means the graph is symmetric with respect to the origin.
That in turn means that f(0) = 0.

(2) f''(x) is positive for all x > 0.

That means the graph is concave up for all x > 0.
The fact that f(x) is odd then means the graph is concave down for all x < 0.

(3) f'(0) is negative.

That means the slope of the graph at x=0 is negative (downhill to the right).

(4) f'(1) = 0.

That means the slope of the graph is 0 (the tangent to the graph is horizontal) at x=1.
That in turn means the slope is also 0 at x = -1.

(5) f(2) = 0.

That means that f(-2) = 0 also.

I can't draw a graph of such a function, because no kind of graph that I know how to graph with the tools on this forum has all those characteristics.

The graph below of the polynomial almost works, except that the slopes of 0 are not at exactly x=1 and x=-1.