Problem 85 - A Complete Resource Book in Mathematics for JEE Main

Question

The second drivative \(f^{\prime \prime}(x)\) of the function \(f(x)\) exists for all \(x\) in \([0,1]\) and satisfies \(\left|f^{\prime \prime}(x)\right| \leq 1 .\) If \(f(0)=f(1)\), then for all \(x\) in \([0,1]\) (A) \(\left|f^{\prime}(x)\right|<1\) (B) \(\left|f^{\prime}(x)\right|>1\) (C) \(\left|f^{\prime}(x)\right|=1\) (D) \(f(x)\) is constant

Step-by-Step Solution

Answer

(A) \(|f'(x)| < 1\) for all \(x\) in \([0,1]\).

Step 1: Analyze Given Condition

The second derivative \(f''(x)\) exists and satisfies \(|f''(x)| \leq 1\) for all \(x\) in \([0,1]\). This means the rate of change of the first derivative \(f'(x)\) with respect to \(x\) is never more than 1 in magnitude.

Step 2: Use Mean Value Theorem

By the Mean Value Theorem, there exists a point \(c\) in \((0, 1)\) such that \(f'(c) = \frac{f(1) - f(0)}{1-0} = \frac{0}{1} = 0\) since \(f(0) = f(1)\).

Step 3: Analyze Implications of the Constraints

Given that \(f'(c) = 0\) and \(|f''(x)| \leq 1\), we know that \(f'(x)\) cannot change by more than 1 over the interval \([0,1]\). Since \(f'(c) = 0\) and the maximum possible change over the interval (due to the second derivative constraint) is up to 1 unit, this implies that \(|f'(x)| < 1\) for all \(x\).

Step 4: Conclusion

As the derivative \(f'(x)\) is bounded such that \(|f'(x)| < 1\) for all \(x\) in \([0,1]\), the correct answer is that the derivative is always less than 1 in magnitude.

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental concept in calculus that offers valuable insights into how functions behave. In essence, this theorem states that for a function that is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), there exists at least one point \(c\) within that interval where the derivative of the function \(f'(c)\) matches the average rate of change over \([a, b]\).
This is mathematically represented as:

• \[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

In practice, the theorem indicates that at some point, the instantaneous rate of change (slope of the tangent) is equal to the average rate of change across the interval.
For our exercise, since \(f(0) = f(1)\), the average rate of change is zero. Therefore, according to the theorem, there exists a point \(c\) in \((0, 1)\) where \(f'(c) = 0\). This concept helps us conclude that somewhere within our interval, the slope of the function is flat, which, combined with derivative constraints, indicates stable behavior of the function.

Second Derivative

The second derivative, denoted as \(f''(x)\), provides information about the concavity and the rate at which the slope of a function is changing.
In other words, while the first derivative \(f'(x)\) tells us about the slope of the function at a point—whether it is increasing or decreasing—the second derivative gives us further insight into how this slope is evolving:

• If \(f''(x) > 0\), the function is concave up (shaped like a cup), indicating the slope is increasing.
• If \(f''(x) < 0\), the function is concave down (shaped like a cap), suggesting the slope is decreasing.

In the given problem, the second derivative constraint \(|f''(x)| \leq 1\) ensures that the rate of change of \(f'(x)\) does not exceed 1 in magnitude over the interval \([0, 1]\).
This means the change in slope is controlled and cannot become too steep. Since \(f'(c) = 0\) at some point due to mean value theorem, coupled with \(f''(x)\)'s constraint, it implies \(f'(x)\) varies smoothly without exceeding a slope of 1 in absolute terms, resulting in consistent behavior across the interval.

Derivative Constraints

Derivative constraints refer to limitations placed on the derivatives of a function, affecting its behavior.
In this exercise, the second derivative constraint \(|f''(x)| \leq 1\) regulates how rapidly the slope of \(f(x)\) can change on the interval \([0, 1]\).

• The restriction implies that the first derivative \(f'(x)\) cannot experience drastic changes.
• This constraint ensures the slope remains within certain bounds, preventing extreme upward or downward slopes.

Having \(f(0) = f(1)\) further plays a significant role. It implies that the average change over the interval \([0, 1]\) is zero.
With \(f'(c) = 0\) at some point, \(f'(x)\) has to stay within \(-1\) and \(1\) because \(f''(x)\) limits how it changes. Therefore, for all points \(x\) between \(0\) and \(1\), the restraint enforced by \(|f'(x)| < 1\) is adhered to, ensuring that the derivative remains smooth and constrained within these bounds, resulting in the function demonstrating consistent properties over the range.