DerivativeApplications

# 8. inflection points and curvature

To study the curvature of a derivable function, you have to find the intervals in which the function is concave up or concave down.

f is **concave up** in c if the graph is above the tangent line to the curve in c.

f is **concave down** in c if the graph is under the tangent line to the curve in c.

An

**inflection point**is a point on a curve at which the curvature or concavity changes.If f is derivable in (a,b)

– f is concave up in (a,b) ↔ f’’(x) > 0

– f is concave down in (a,b) ↔ f’’(x) < 0

If is derivable in cЄ

**R**, then:- f has an inflection point in c → f’’(c) = 0

**Exercise**: study the curvature of the function y = x·e^{x} and find its inflection points

Solutions: inflection point (-2, -2/e^{2}); concave up (-2,∞); concave down (-∞,-2)

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