Problem 96 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

For all twice differentiable functios \(f: \mathrm{R} \rightarrow \mathrm{R}\), with \(f(0)=f(1)=f^{\prime}(0)=0\) (a) \(f^{\prime \prime}(x) \neq 0\) at every point \(x \in(0,1)\) (b) \(f^{\prime \prime}(x)=0\), for some \(x \in(0,1)\) (c) \(f^{\prime \prime}(0)=0\) (d) \(f^{\prime \prime}(x)=0\), at every point \(x \in(0,1)\)

Step-by-Step Solution

Answer

Option (b) is correct: \(f''(x) = 0\) for some \(x \in (0,1)\).

Step 1: Understand the Given Constraints

We have a function \( f \) such that \( f(0) = f(1) = f'(0) = 0 \). This implies that the function is zero at both endpoints and has a horizontal tangent at zero, meaning it flattens out at \( x = 0 \).

Step 2: Analyze f(x) Based on Initial Conditions

Since \( f(0) = f(1) = 0 \), \( f(x) \) has zero value at \( x = 0 \) and \( x = 1 \). \( f'(0) = 0 \) indicates that the slope of the tangent to the graph is zero at \( x = 0 \). This suggests there's a turning point or an inflection at \( x = 0 \).

Step 3: Consider the Second Derivative f''(x)

To investigate options (a), (b), (c), and (d), we explore the behavior of \( f''(x) \). Because \( f'(0) = 0 \) and given we need conditions on \( f''(x) \), the point where \( f'(0) = 0 \) may relate to a local maximum, minimum, or inflection.

Step 4: Test Each Option

(a) \( f''(x) eq 0 \) for all \( x \in (0,1) \) implies \( f \) cannot flatten completely between \( (0,1) \). Given \( f' \) might change on \( (0,1) \), this is unlikely.(b) \( f''(x) = 0 \) for some \( x \in (0,1) \) suggests an inflection point, which is plausible given \( f \) could change concavity.(c) \( f''(0) = 0\) is plausible due to \( f'(0) = 0 \), indicating a possible inflection.(d) \( f''(x) = 0 \) everywhere between \((0,1)\) would mean constant slope, contradicting \( f(0) = f(1) = 0 \).

Step 5: Conclusion

Option (b) \( f''(x) = 0 \) for some \( x \in (0,1) \) is correct. Given the constraints and potential behavior of \( f \), it is reasonable to expect an inflection point within the open interval \((0,1)\).

Key Concepts

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), provides information about the concavity of the function as well as the acceleration of the function's rate of change. Here’s how it works:

• If \( f''(x) > 0 \) at a certain point, the function is concave up, often resembling a cup shape at that point.
• If \( f''(x) < 0 \), the function is concave down, similar to an upside-down cup at that location.
• If \( f''(x) = 0 \), it might indicate an inflection point where the function changes concavity.

Understanding the behavior of the second derivative helps in predicting where the graph bends or changes its bending direction. In problems dealing with twice differentiable functions, the second derivative plays a key role in determining various characteristics of the function, including the presence of inflection points or local extrema.

Inflection Points

An inflection point occurs where a function changes its concavity. This means the curve of the graph transitions from concave up to concave down, or vice versa. Inflection points are detected through the second derivative.

• If \( f''(x) = 0 \) and there is a change in sign for \( f''(x) \) as \( x \) crosses a certain point, that point is an inflection point.
• Unlike turning points, inflection points are not necessarily peaks or valleys, but places where the curve changes its bending direction.

Identifying inflection points is crucial when analyzing the curvature and behavior of a function since these points give insight into where and how the function's slope accelerates or decelerates.

Differential Calculus

Differential calculus is a branch of mathematics that focuses on rates of change and slopes. It deals primarily with derivatives, which are used to calculate how a function is changing at any given point.

• The first derivative, \( f'(x) \), provides the slope or the rate of change of the function. It tells us how steep a graph is at a point.
• The second derivative, \( f''(x) \), reveals acceleration, showing how the rate of change itself is evolving.

Using differential calculus, mathematicians can solve problems related to finding maximum and minimum values of functions, determining rates, and analyzing motion. It is a critical skill for understanding the dynamics of varying quantities and is applicable in fields such as physics, engineering, and economics.