上凹还是下凹,叫法相反的凹函数 Concave Upward and Downward
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Concave Upward and Downward
Concave upward is when the slope increases: Concave downward is when the slope decreases:What about when the slope stays the same (straight line)? It could be both! See footnote.
Here are some more examples:
Concave Downward is also called Concave or Convex Upward
Concave Upward is also called Convex or Convex Downward
Finding where ...
Usually our task is to find where a curve is concave upward or concave downward:
Definition
The key point is that a line drawn between any two points on the curve won't cross over the curve:
Let's make a formula for that!
First, the line: take any two different values a and b (in the interval we are looking at):
Then "slide" between a and b using a value t (which is from 0 to 1):
x = ta + (1−t)b
- When t=0 we get x = 0a+1b = b
- When t=1 we get x = 1a+0b = a
- When t is between 0 and 1 we get values between a and b
Now work out the heights at that x-value:
When x = ta + (1−t)b:
- The curve is at y = f( ta + (1−t)b )
- The line is at y = tf(a) + (1−t)f(b)
And (for concave upward) the line should not be below the curve:
For concave downward the line should not be above the curve (≤ becomes ≥):
And those are the actual definitions of concave upward and concave downward.
Remembering
Which way is which? Think:
Concave Upwards = CUP
Calculus
Derivatives can help! The derivative of a function gives the slope.
- When the slope continually increases, the function is concave upward.
- When the slope continually decreases, the function is concave downward.
Taking the second derivative actually tells us if the slope continually increases or decreases.
- When the second derivative is positive, the function is concave upward.
- When the second derivative is negative, the function is concave downward.
Footnote: Slope Stays the Same
What about when the slope stays the same (straight line)?
Saying Strictly Concave upward or Strictly Concave downward means a straight line is not OK.
Otherwise a straight line is acceptable for both.
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