Concave

Concave upward is when the slope increases:

Concave downward is when the slope decreases:

Info

Concave Upward is also called Convex, or sometimes Convex Downward Concave Downward is also called Concave, or sometimes Convex Upward

Determine Concave

• If $f^{\prime \prime }(x)>0$ for a given interval, the function is concave up within that interval.
• If $f^{\prime \prime }(x)<0$ for a given interval, the function is concave down within that interval.

Example

Let's consider the function f(x)= x^3 − 3x.

• The first derivative is $y^{\prime }=3x^2-3$
• The second derivative is $y^{\prime \prime }=6x$

Next, we’ll set the second derivative equal to zero to find the Inflection Points: $6x=0$

Solving for $x$: $x=0$

• f(x) is concave downward up to x = 0
• f(x) is concave upward from x = 0