Question

Let x₀ be a point in the domain of definition of a real valued function ${\displaystyle f}$ 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.

Options

• ${\displaystyle f}$ is increasing at x₀ if x₁ < x₂ in I ⇒ ${\displaystyle f\left(x_1\right)\le f\left(x_2\right)}$.

• ${\displaystyle f}$ is strictly increasing at x₀ if x₁ < x₂ in I ⇒ ${\displaystyle f\left(x_1\right)<f\left(x_2\right)}$.

• ${\displaystyle f}$ is decreasing at x₀ if x₁ < x₂ in I ⇒ ${\displaystyle f\left(x_1\right)\ge f\left(x_2\right)}$.

• All the above

Answer 1

All the above

Explanation:

Let ${\displaystyle f}$ be a function on [a, b] and differentiable in an open interval (a, b) then

(i) ${\displaystyle f}$ is increasing on [a, b] if ${\displaystyle f^{\prime }\left(x\right)>0}$ for each x ∈ (a, b)

(ii) ${\displaystyle f}$ is decreasing on [a, b] if ${\displaystyle f^{\prime }\left(x\right)<0}$ for each x ∈ (a, b)

(iii) ${\displaystyle f}$ is constant on [a, b] if ${\displaystyle f^{\prime }\left(x\right)=0}$ for each x ∈ (a, b)