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).