CONTINUOUS FUNCTION WITH F' & GRAPH - CALCULUS

Calculus
Tutor: None Selected Time limit: 1 Day

draw a possible graph of a continuous function y=f(x) that satisfies the three conditions: I. f'(x)>0 for 1<x<3 II. f'(x)<0 for x<1 and x>3 III. f'(x)=0 for x=1 and x=3

Sep 30th, 2015

I'm glad to help!

What you need to do is draw what's called a "sign chart." Draw a straight, horizontal line and label it as f'(x) from x= -∞ to x= +∞. Then mark the midpoint on the line as x= 0 and mark x= 1 and x = 3 on the right. On top of the line, put a minus sign from -∞ to 1, a plus sign from 1 to 3, and from 3 to +∞; these are the signs of the derivative f'(x), which is the rate of change of f(x). For positive values of f'(x), the graph of f(x) is increasing. For negative values of f'(x), the graph of f(x) is decreasing. For f'(x) = 0, the graph of f(x) is neither increasing nor decreasing; its tangent line is horizontal (parallel with the x-axis). This happens at x= 1 and x= 3 (even though the graph of f(x) is increasing at all x-values between 1 and 3 and all x-values after 3, it is not increasing or decreasing at x= 1 or x=3). 

Lastly, the values of f(x) don't matter since you're only given the sign of f'(x); the actual y-values of f(x) don't matter, just the change in y with respect to infinitely small change in x (dy/dx). You can draw the graph between any y-values that you want. 

Ask for clarification if necessary!
Sep 30th, 2015

Oh, and the graph is continuous, so it can't have any discontinuities or "kinks" in it; it must be smooth and have all points connected. The graph of f(x) should look somewhat like an S turned counterclockwise 90°.

Sep 30th, 2015

Correction! The graph is decreasing for all x-values greater than 3 because f'(x>3) is negative. Disregard the second part of the last parentheses text in the first reply. 

Sep 30th, 2015

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Sep 30th, 2015
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Sep 30th, 2015
Dec 6th, 2016
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