Suppose f(1) = 5 and f'(1) = 0. What can we conclude of (1,5) if...?

[mika0]

how to do this?

17. Suppose f(1) = 5 and f'(1) = 0. What can we conclude about the point (1, 5) if:

(a) f'(x) < 0 for x < 1, and f'(x) > 0 for x > 1?
(b) f'(x) < 0 for x < 1, and f'(x) < 0 for x > 1?
(c) f'(x) > 0 for x < 1, and f'(x) < 0 for x > 1?
(d) f'(x) > 0 for x < 1, and f'(x) > 0 for x > 1?

 

[ksdhart]

I'd begin by drawing a rough graph of one possible function for each of the four scenarios. Let's start with the first one. When x < 1, the derivative is negative. What does that tell you about the graph of the function? And when x > 1, the derivative of positive. What does that tell you about the graph of the function? Now that you've drawn a sketch of what the graph might look like, what can you say about the point (1, 5)? Repeat these steps for the other three scenarios.

If you get stuck again, that's okay. But when you reply back, please include all of your work, even if you know it's wrong.

 

[stapel]

17. Suppose f (1) = 5 and f '(1) = 0. What can we conclude about the point (1, 5) if:

(a) f '(x) < 0 for x < 1, and f '(x) > 0 for x > 1?
(b) f '(x) < 0 for x < 1, and f '(x) < 0 for x > 1?
(c) f '(x) > 0 for x < 1, and f '(x) < 0 for x > 1?
(d) f '(x) > 0 for x < 1, and f '(x) > 0 for x > 1?

how to do this?

If the slope at a point (that is, if the derivative) is equal to zero, what does this tell you about the "shape" or "angle" of the graph at that point?

If the derivative is negative, what does that say? If it is positive, what does that say?

How does the shape of a graph "near" a critical point relate to the sign of the derivative on either side of that critical point?