Concave Curve Concavity Test:
(i) If ƒ"(x) > 0 V x ? [a, b], then the curve y = ƒ(x) is concave upward on [a, b].
(ii) If ƒ"(x) < 0 V x ? [a, b], then the curve y = ƒ(x) is concave downward on [a, b].
Proof: (i) Let ƒ"(x) > 0 V x ? [a, b].
Let P (x0, ƒ(x0)) be such point on the curve y = ƒ(x). The equation of the tangent on the curve at P is
y - ƒ(x0) = ƒ(x0) (x - x0)
i.e. y = ƒ(x0) + ƒ'(x0) (x - x0)
y is the coordinate of any arbitrary point on the tangent line. Let A (x, ƒ(x)) be a variable point on the provided curve. Let the coordinate at A intersect the tangent line at A'.
If AA' = Ø (x), then
Ø (x) = ƒ(x) - [ƒ(x0) + ƒ'(x0) (x - x0)]
=> Ø' (x) = ƒ'(x) - ƒ(x0)
And, Ø" (x) = ƒ"(x)
It follows from relations that
Ø (x0) = 0, Ø' (x0) = 0
And Ø" (x0) = ƒ"(x0)
Since Ø (x0) = 0 and Ø" (x0) > 0 (? ƒ"(x0), Ø (x) has a minimum at x = x0)
Therefore ∃ a δ > 0 such that Ø (x) > Ø (x0) in ]x0 - δ, x0 + δ[, x ≠ x0
i.e. Ø (x) > 0, which seems that A lies above the tangent at P.
Therefore the curve y = ƒ(x) is concave upward on [a, b].
Same as we can prove the second part of the theorem.