Guide To Determining Concavity With Second Derivative: Step-By-Step Explanation.

Determine Concavity

f is concave up: f’ is increasing: f” > 0 (positive)f is concave down: f’ is decreasing: f” < 0 (negative)

Concavity is a term used to describe the shape of a curve. It refers to whether the curve is bending upwards (concave up) or downwards (concave down). The concavity of a curve can be determined by examining its second derivative.

To determine concavity, follow these steps:

1. Take the second derivative of the curve: f”(x)
2. Identify the critical points of the second derivative: where f”(x) = 0 or undefined
3. Evaluate the second derivative at each of the critical points:
– If f”(x) > 0, the curve is concave up
– If f”(x) < 0, the curve is concave down If the second derivative is positive, the curve is concave upward, meaning the curve is curving upward. This is like a smiley face. If the second derivative is negative, the curve is concave downward, which is opposite in nature, meaning the curve is curving downward. This shape is like a frowny (sad) face. Note that, if the second derivative is zero at a critical point, you cannot determine the concavity of the curve at that point. You need to evaluate the second derivative on either side of the critical point to determine the concavity. Remember that concavity determines a curve's directional changes, maxima, and minima.

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