Question

Let \( f(x) \) be a nonnegative differentiable function on \( [a, b] \subset \mathbb{R} \) such that \( f(a) = 0 = f(b) \) and \( |f'(x)| \leq 4 \). Let \( L_1 \) and \( L_2 \) be the straight lines given by the equations \[ y = 4(x - a) \quad \text{and} \quad y = -4(x - b), \text{ respectively. Then which of the following statements is (are) TRUE?} \]

• A. The curve \( y = f(x) \) will always lie below the lines \( L_1 \) and \( L_2 \)
• B. The curve \( y = f(x) \) will always lie above the lines \( L_1 \) and \( L_2 \)
• C. \( \int_a^b f(x) dx \leq (b - a)^2 \)
• D. The point of intersection of the lines \( L_1 \) and \( L_2 \) lie on the curve \( y = f(x) \)

Hint:

When dealing with bounded derivative conditions, the integral of the function is bounded by the areas under the bounding lines.

Answer 1

The Correct Option is A, C

Step 1: Analyze the function.
Since \( |f'(x)| \leq 4 \), this means the slope of \( f(x) \) is bounded, and \( f(x) \) cannot grow faster than the straight lines \( L_1 \) and \( L_2 \). Therefore, the function \( f(x) \) lies between these two lines.
Step 2: Analyzing the options.
- (A) The curve \( y = f(x) \) will always lie below the lines \( L_1 \) and \( L_2 \): This is true because the slope of \( f(x) \) is bounded, and the curve will never exceed the bounds set by these lines.
- (B) The curve \( y = f(x) \) will always lie above the lines \( L_1 \) and \( L_2 \): This is false because the function could lie below these lines at certain points.
- (C) \( \int_a^b f(x) dx \leq (b - a)^2 \): This is correct because the area under the curve \( f(x) \) is bounded by the area under the lines \( L_1 \) and \( L_2 \), which is \( (b - a)^2 \).
- (D) The point of intersection of the lines \( L_1 \) and \( L_2 \) lies on the curve \( y = f(x) \): This is false, as the function does not necessarily intersect at the same point.
Step 3: Conclusion.
The correct answers are (A) and (C).