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:

 

At x = ta + (1−t)b:

  • The curve will be at y = f( ta + (1−t)b )
  • The line will be 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.

Example: the function x2

Its derivative is 2x (see Derivative Rules)

2x continually increases, so the function is concave upward.

Its second derivative is 2

2 is positive, so the function is concave upward.

Both give the correct answer.

 

Example: f(x) = 5x3 + 2x2 − 3x

Let's work out the second derivative:

  • The derivative is f'(x) = 15x2 + 4x − 3 (using Power Rule)
  • The second derivative is f''(x) = 30x + 4 (using Power Rule)

 

And 30x + 4 is negative up to x = −4/30 = −2/15, and positive from there onwards. So:

f(x) is concave downward up to x = −2/15
f(x) is concave upward from x = −2/15 on

 

Note: The point where it changes is called an inflection point.

 

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.

Example: y = 2x + 1

2x + 1 is a straight line.

 

It is Concave upward.
It is also Concave downward.

It is not Strictly Concave upward.
And it is not Strictly Concave downward.