**MAXIMA AND MINIMA PROBLEMS**

1. Without using the derivative, show that f(x) = (½)^{x} is strictly decreasing on R.

**Solution: **Given that f(x) = (½)^{x}

Let x₁, x₂ ϵ R

Let x₁ < x₂ ⇒ (½)^{x}^{₁} > (½)^{x}^{₂}

⇒ f(x₁) > f(x₂).

∴ f is strictly decreasing on R.

2. Without using the derivative, show that f(x) = 3x + 7 is strictly decreasing on R

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

Let x₁, x₂ ϵ R

Let x₁ < x₂ 3x₁ < 3x₂

⇒ 3x₁ + 7 < 3x₂ + 7

⇒ f(x₁) < f(x₂)

⇒ f is strictly decreasing on R

3. Show that x+1/x is increasing on [1, ∞)

**Solution: **f(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

4. State the point at which the following functions are increasing and the points at which they are decreasing f(x) = x³ – 3x²

**Solution: **f(x) – x³ – 3x²

f’(x) = 3x² – 6x = 3x (x-2)

f(x) is increasing function f’(x) > 0

3x (x – 2) > 0

Therefore x does not lie between 0 and 2

Therefore f(x) is increasing in (-∞, 0) È (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)