**Maxima and Minima –
Problems**

**Example 1: Show that f(x) = 3x + 7 is
strictly increasing on R. f(x) = 3x + 7**

**Solution: **Given that f(x) = 3x + 7

let x₁ x₂ ϵ R

let x₁ < x₂

3x₁ < 3x₂

3 x₁ + 7 < 3x₂ + 7

f(x₁) < f(x₂)

Therefore, above function is strictly increasing

**Example 2: Show that x + 1/x is
increasing on [1, ∞).**

**Solution: **Given thatf(x) = x + 1/x

f’(x) = 1 – 1/x²

since x ϵ [1, ∞), 1- 1/x² > 0

f’(x) > 0

Therefore, f(x) is increasing function

**Example 3: State the point at which
the following function (from 1 to 4) are increasing and the point at which they
are decreasing** **f(x) = x³ – 3x²**

**Solution: **Given that f(x) = x³ – 3x²

f’(x) = 3x² – 6x

= 3x (x – 2)

f(x) is increasing if f’(x) > 0

3x (x – 2) > 0

X does not lie between 0 and 2

f(x) is increasing in (-∞, 0) u (2, ∞)

f(x) is decreasing if f’(x) < 0

x (x – 2) < 0

x lies between 0 and 2

f(x) is decreasing in (0, 2).