Graphs of functions and their derivatives

[thestranger]

The graph of f′ (not f) is given below.

​

This is what I think
At which of the marked values of x is
A. f(x) greatest? x= 3
B. f(x) least? x= 5
C. f′(x) greatest? x= 1
D. f′(x) least? x= 6
E. f″(x) greatest? x= 3
F. f″(x) least? x=4

 

[ksdhart]

Can you please show us what steps, if any, you've already attempted? If we know what you've tried (even if you know for sure your answer is wrong, or incomplete, include it anyway), it will help us know how to help you). Also, please note that this forum is called "Free Math Help," and not "We do your homework for you".

If you need a hint to get started, think about what you know about the behavior of the function based on the values of the derivative at a given point.

 

[stapel]

The graph of f′ (not f) is given below.



At which of the marked values of x is {This is what I think}
A. f(x) greatest? {x = 3}
B. f(x) least? {x = 5}
C. f'(x) greatest? {x = 1}
D. f'(x) least? {x = 6}
E. f"(x) greatest? {x = 3}
F. f"(x) least? {x = 4}

First, the x-values are named x₁, x₂, x₃, x₄, x₅, and x₆. These are clearly not the numerical vales of the x-vales; for instance, x₁, being to the left of the y-axis, cannot possibly be equal to a positive 1. :shock:

For (C) and (D), on what basis did you come to your conclusions? The graph displays f', so there isn't even any "thinking" required; you need only copy down the name for the line leading to (C) the highest (marked) point on the blue line and to (D) the lowest (marked) point on the blue line. The line for x₂ leads to a higher point than does the line for x₁, so why did you choose x₁?

And so forth, for each of your answers. Please be complete. Thank you!

 

[Steven G]

The graph of f′ (not f) is given below.

​

This is what I think
At which of the marked values of x is
A. f(x) greatest? x= 3
B. f(x) least? x= 5
C. f′(x) greatest? x= 1
D. f′(x) least? x= 6
E. f″(x) greatest? x= 3
F. f″(x) least? x=4

Based on f '(x), you should sketch f(x) and f"(x) and only then show us your work. Thanks.

 

[Ishuda]

First, a comment. The is no value indicated for the vertical axis but all that is really needed is the 'zero line' which I will assume is where the lower end of the vertical dotted lines end. Now, on to the problem.

To provide a little more of a hint than ksdhart gave:
For the questions about f; what is the behavior of a function if its derivative is always positive?

For the questions about f'; just read them off the graph.

For the questions about f''; Since f'' is the derivative of f' and the derivative represents the slope, what can you say about the slope of the graph?