Question

Let \( f(x) \) be a continuous function on \([a, b]\) and differentiable on \((a, b)\). Then, this function \( f(x) \) is strictly increasing in \((a, b)\) if:

• A. \( f'(x) <0, \, \forall x \in (a, b) \)
• B. \( f'(x)>0, \, \forall x \in (a, b) \)
• C. \( f'(x) = 0, \, \forall x \in (a, b) \)
• D. \( f(x)>0, \, \forall x \in (a, b) \)

Hint:

A strictly increasing function has a positive derivative throughout the interval. A derivative equal to zero implies a constant function.

Answer 1

The Correct Option is B

Step 1: Understand the condition for increasing functions. A function \( f(x) \) is strictly increasing on an interval \((a, b)\) if for any \( x_1, x_2 \in (a, b) \), where \( x_1 <x_2 \), we have: \[ f(x_1) <f(x_2) \] Step 2: Apply the derivative test. The derivative \( f'(x) \) gives the slope of the tangent to the curve. If \( f'(x)>0 \) for all \( x \in (a, b) \), the slope is positive, which means the function is strictly increasing. Final Answer: \[ \boxed{\text{(b) } f'(x)>0, \, \forall x \in (a, b)} \]