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Μάθημα: Διαφορικός Λογισμός > Ενότητα 5

Μάθημα 10: Connecting f, f', and f''
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Διαφορικός Λογισμός
Analyzing functions
Connecting f, f', and f''
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"Calculus-based reasoning" with the second derivative of a function can be used to justify claims about the concavity of the original function and about its inflection points.
We've learned that the first derivative f′ gives us information about where the original function f increases or decreases, and about where f has extremum points.
The second derivative f″ gives us information about the concavity of the original function f and about where f has inflection points.

Let's review what concavity is all about.

A function is concave up when its slope is increasing. Graphically, a graph that's concave up has a cup shape, ∪.
The graph of f is concave up (notice its ∪ shape). Notice how, as x increases, the slope is increasing.
Similarly, a function is concave down when its slope is decreasing. Graphically, a graph that's concave down has a cap shape, ∩.
The graph of g is concave down (notice its ∩ shape). Notice how, as x increases, the slope is decreasing.
A point of inflection is where a function changes concavity.

How f″ informs us about the concavity of f

When the second derivative f″ is positive, that means the first derivative f′ is increasing, which means that f is concave up. Similarly, a negative f″ means f′ is decreasing and f is concave down.
f″f′f
positive +increasing ↗concave up ∪
negative −decreasing ↘concave down ∩
crossing x-axis (changes sign)extremum point (changes direction)inflection point (changes concavity)
Here’s a graphical example:
f″f′f
Notice how f is concave down to the left of x=c and concave up to the right of x=c.
Πρόβλημα 1
Let f be a twice differentiable function. This is the graph of its second derivative, f″.
Over which interval is f always concave up?
Επιλέξτε 1 απάντηση:

Common mistake: Confusing the relationship between f, f′, and f″

Remember that for f to be concave up, f′ needs to be increasing and f″ needs to be positive. Other behaviors of f, f′, and f″ aren't necessarily related.
For example, in Problem 1 above, f″ is concave up over the interval [−8,−2] but it doesn't mean f is concave up on that interval.
Πρόβλημα 2
Let h be a twice differentiable function. This is the graph of its second derivative, h″.
Where does h have an inflection point?
Επιλέξτε 1 απάντηση:

Want more practice? Try this exercise.

Common mistake: Misinterpreting the graphical information presented

Imagine a student solving Problem 2 above, thinking that the graph is of the first derivative of h. In that case, h would have an inflection point at A and B, because these are the points where h′ changes its direction. This student would be wrong, because this is the graph of the second derivative and the correct answer is D.
Remember to always make sure you understand the information given. Are we given the graph of the function f, the first derivative f′, or the second derivative f″?
Πρόβλημα 3
The twice differentiable function g and its second derivative g″ are graphed.
Four students were asked to give an appropriate calculus-based justification for the fact that g has an inflection point at x=−2.
Can you match the teacher's comments to the justifications?
1

Using the second derivative to determine whether an extremum point is a min or a max

Imagine we are given that a function f has an extremum point at x=1, and that it's concave up over the interval [0,2]. Can we tell, based on this information, whether that extremum point is a minimum or a maximum?
The answer is YES. Recall that a function that's concave up has a cup ∪ shape. In that shape, a curve can only have a minimum point.
Similarly, if a function is concave down when it has an extremum, that extremum must be a maximum point.
Πρόβλημα 4
The twice differentiable function h and its second derivative h″ are graphed.
Given that h′(−4)=0, what is an appropriate calculus-based justification for the fact that h has a relative maximum at x=−4?
Επιλέξτε 1 απάντηση:

Want more practice? Try this exercise.

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