**(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 (x _{0}, ƒ(x_{0})) **be any point on the curve

**y = ƒ(x)**. The equation of the tangent to the curve at

**P**is

**y – ƒ(x _{0}) = ƒ(x_{0}) (x – x_{0})**

i.e. **y = ƒ(x _{0}) + ƒ’(x_{0}) (x – x_{0})**

**y **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(x _{0}) + ƒ’(x_{0}) (x – x_{0})]**

**Ø’ (x) = ƒ’(x) – ƒ(x _{0}) **

And, **Ø” (x) = ƒ”(x)**

It follows from these relations that

**Ø (x _{0}) = 0, Ø’ (x_{0}) = 0**

And **Ø” (x _{0}) = ƒ”(x_{0})**

Since **Ø (x _{0}) = 0 **and

**Ø” (x**has a minimum at

_{0}) > 0 (∵ ƒ”(x_{0}), Ø (x)**x = x**

_{0})Thus **∃ a δ > 0** such that **Ø (x) > Ø (x _{0}) **in

**]x**

_{0}– δ, x_{0}+ δ[, x ≠ x_{0}

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.