Concave Upwards and Downward

What most when the gradient stays the same (directly line)? Information technology could be both! Meet footnote.

Hither are some more examples:

concave upward and downward examples

Concave Upward is also chosen Convex, or sometimes Convex Downwards

Concave Downward is likewise called Concave, or sometimes Convex Upward

Finding where ...

Usually our task is to notice where a curve is concave upward or concave downwardly:


concave sections

Definition

A line drawn between any two points on the bend won't cantankerous over the curve:

concave upward yes and no examples

Let's make a formula for that!

First, the line: take any 2 different values a and b (in the interval we are looking at):

concave upward between a and b

And so "slide" between a and b using a value t (which is from 0 to ane):

x = ta + (1−t)b

  • When t=0 nosotros become x = 0a+1b = b
  • When t=i we get x = 1a+0b = a
  • When t is betwixt 0 and 1 nosotros go values between a and b

Now work out the heights at that ten-value:

concave line t

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 upwardly) the line should not be below the curve:

concave upwnward f( ta + (1-t)b ) <= tf(a) + (1-t)f(b)

For concave downwardly the line should non exist higher up the curve (≤ becomes ≥):

concave downward f( ta + (1-t)b ) >= tf(a) + (1-t)f(b)

And those are the actual definitions of concave upward and concave downwards.

Remembering

Which mode is which? Call back:

concave up: cup
Concave Upwards = Loving cup

Calculus

Derivatives can help! The derivative of a function gives the gradient.

  • When the slope continually increases, the function is concave upward.
  • When the slope continually decreases, the function is concave downward.

Taking the second derivative really tells us if the slope continually increases or decreases.

  • When the second derivative is positive, the part is concave up.
  • When the 2d derivative is negative, the function is concave downward.

Example: the role x2

x^2 concave upward

Its derivative is 2x (run across Derivative Rules)

2x continually increases, and then the function is concave upward.

Its second derivative is 2

ii is positive, so the office is concave upward.

Both give the correct respond.

Case: f(ten) = 5x3 + 2xtwo − 3x

5x^3 + 2x^2 - 3x inflection point

Let'south piece of work out the second derivative:

  • The derivative is f'(x) = 15x2 + 4x − 3 (using Power Rule)
  • The 2nd derivative is f''(x) = 30x + 4 (using Ability Rule)

And 30x + iv is negative up to x = −4/30 = −2/15, and positive from at that place onwards. And then:

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

f(10) is concave upward from x = −two/15 on

Annotation: The bespeak where it changes is chosen an inflection point.

Footnote: Gradient Stays the Same

What nearly when the gradient stays the aforementioned (straight line)?

A straight line is acceptable for concave upwards or concave downward.

Merely when we use the special termsstrictly concave upwardly or strictly concave downward and so a direct line is non OK.

2x+1

Example: y = 2x + 1

2x + 1 is a straight line.

It is concave up.
It is as well concave downward.

It is non strictly concave upwardly.
And it is not strictly concave downward.