Concave Curve Concavity Test

Math Assignment Help

(i) If ƒ”(x) > 0 V x ϵ [a, b], then the curve y = ƒ(x) is concave upward on [a, b].

(ii) If ƒ”(x) < 0 V x ϵ [a, b], then the curve y = ƒ(x) is concave downward on [a, b].

Proof: (i) Let ƒ”(x) > 0 V x ϵ [a, b].

Let P (x0, ƒ(x0)) be any point on the curve y = ƒ(x). The equation of the tangent to the curve at P is

y – ƒ(x0) = ƒ(x0) (x – x0)

i.e. y = ƒ(x0) + ƒ’(x0) (x – x0)

is the ordinate of any point on the tangent line. Let A (x, ƒ(x)) be a variable point on the given curve. Let the ordinate at A cut 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 these 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)

Thus ∃ a δ > 0 such that Ø (x) > Ø (x0in ]x0 – δ, x0 + δ[, x ≠ x0

i.e. Ø (x) > 0, which means that A lies above the tangent at P.

Hence the curve y = ƒ(x) is concave upward on [a, b].

Similarly we can prove the second part of the theorem.

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