Concave or concave, which is called the opposite, concave upward and downward

concave upward and downward

concave upward is when the slope increases:
concave downward is when the slope decreases:

What is the slope stays the same (straight line)? It could be both! See footnote.

Here is 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 was to find where a curve is concave upward or concave downward:

Definition

The key point is, a line drawn between any and points on the curve won ' t cross over the curve:

Let's make a formula for that!

First, the Line:take any of the different values a and b (in the interval we is looking at):

Then "slide" between a and b using a value T (which are from 0 to 1):

x = Ta + (1−t) b when t=0 we get x = 0a+1b = b If 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 of the heights at that X-value:

When x = Ta + (1−t) b: The curve was at y = f (ta + (1−t) b) , the line was at y = tf (a) + (1−t) f (b)

and (for concave upward) The line should isn't be below the curve:

For concave downward , the line should is not being above the curve (≤ becomes ≥):

And those is the actual definitions of concave upward and concave downward. Remembering

Which is Which? Think:

Concave up wards = 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 are positive, the function is concave upward. When the second derivative are negative, the function is concave downward. example:the function X2

Its derivative are 2x (see derivative Rules) 2x continually increases, so the function is concave upward.

Its second derivative are 2 2 are 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 Deriv Ative is F ' (x) = 30x + 4 (using Power Rule)

and 30x + 4 are 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) are concave upward from x =−2/15 on

Note:the point where it changes is called a inflection point.

Footnote:slope stays the same

What is the slope stays the same (straight line)?

Saying strictly concave upward or strictly concave downward means a straight line are not OK.

Otherwise a straight line was 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.

From:http://www.mathsisfun.com/calculus/concave-up-down-convex.html