Concavity and Tangent Functions

In summary, the problem involves proving a statement about a concave down function and its tangent line. The first two lines establish the fact that the function is concave down and the third line shows how this leads to the conclusion that the tangent line must be above the function. However, the last line, which involves a specific value, seems to be causing confusion. The key is to remember that the tangent line must be above the function at all points, including the specific value in question. This leads to the final statement and the proof of the original statement.
  • #1
Strand9202
10
2
Homework Statement
The problem is below. I was asked to explain what is meant in the circled part.
Relevant Equations
None
Here is the problem (8b). I was asked to write out why the circled part was true.
63658155135__B50E8DE7-92CD-4EA2-B76E-74436AD82853.jpeg


I know that since the function is concave down then f"(x)<0. That is a fact. What I am having trouble with is why they can say the next part.

What I thought was L(x) is the tangent line and all tangent lines are above a concave down function, but not sure that is correct or true.
 
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  • #2
I guess I am just lost on the last line, because I know the first 2 lines are true because of concave down.
 
  • #3
It's a simple mistake.
$$L(x)\ge f(x) \quad \Rightarrow\quad L(8) \ge f(8)\quad \Rightarrow \quad f(8)\le L(8) = 1 $$q.e.d.
 

FAQ: Concavity and Tangent Functions

What is a concave function?

A concave function is a function whose graph curves downward, meaning that the function is decreasing at an increasing rate. Mathematically, a function is concave if its second derivative is negative.

How is concavity related to the second derivative?

The second derivative of a function represents the rate of change of the first derivative. If the second derivative is negative, the first derivative is decreasing, indicating a concave function. If the second derivative is positive, the first derivative is increasing, indicating a convex function.

What is the difference between a concave and convex function?

A concave function curves downward, while a convex function curves upward. This means that a concave function is decreasing at an increasing rate, while a convex function is increasing at an increasing rate.

How do you find the points of inflection for a concave function?

The points of inflection for a concave function are where the concavity changes from upward to downward, or vice versa. These points can be found by setting the second derivative equal to 0 and solving for the x-values.

What is the relationship between tangent functions and concavity?

The tangent function is a trigonometric function that represents the ratio of the opposite side to the adjacent side of a right triangle. The tangent function can also be used to describe the slope of a curve at a specific point. In concave functions, the slope of the tangent line decreases as the x-value increases, while in convex functions, the slope of the tangent line increases as the x-value increases.

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