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How Do You Determine Concavity?

James H. Sterling • • 7 min read
How Do You Determine Concavity?

To find when a function is concave, you must first take the 2nd derivative

2nd derivative
The second derivative of a function f can be used to determine the concavity of the graph of f. A function whose second derivative is positive will be concave up (also referred to as convex), meaning that the tangent line will lie below the graph of the function.
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, then set it equal to 0, and then find between which zero values the function is negative. Now test values on all sides of these to find when the function is negative, and therefore decreasing.

How do you know if something is concave up or down?

Taking the second derivative actually tells us if the slope continually increases or decreases.
  1. When the second derivative is positive, the function is concave upward.
  2. When the second derivative is negative, the function is concave downward.

How do you find the concavity?

We can calculate the second derivative to determine the concavity of the function's curve at any point.
  1. Calculate the second derivative.
  2. Substitute the value of x.
  3. If f "(x) > 0, the graph is concave upward at that value of x.
  4. If f "(x) = 0, the graph may have a point of inflection at that value of x.
James H. Sterling
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James H. Sterling

James Sterling reports on renewable energy developments, climate policy, ecological conservation, and green tech innovations around the globe.