Question

Let x₀ be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x₀ –  h, ro + h) containing x₀. Then which of the following statement is/ are true for the above statement.

विकल्प

• `f` is increasing at x₀ if x₁ < x₂ in I ⇒ `f(x_1) ≤ f(x_2)`.

• `f` is strictly increasing at x₀ if x₁ < x₂ in I ⇒ `f(x_1) < f(x_2)`.

• `f` is decreasing at x₀ if x₁ < x₂ in I ⇒ `f(x_1) ≥ f(x_2)`.

• All the above

Answer 1

All the above

Explanation:

Let `f` be a function on [a, b] and differentiable in an open interval (a, b) then

(i) `f` is increasing on [a, b] if `f^'(x) > 0` for each x ∈ (a, b)

(ii) `f` is decreasing on [a, b] if `f^'(x) < 0` for each x ∈ (a, b)

(iii) `f` is constant on [a, b] if `f^'(x) = 0` for each x ∈ (a, b)