Problem 71 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f^{\prime \prime}(x)<0 \forall x \in(a, b)\), then \(f^{\prime}(x)=0\) (A) exactly once in \((a, b)\) (B) at most once in \((a, b)\) (C) at least once in \((a, b)\) (D) None of these

Step-by-Step Solution

Answer

(B) at most once in \((a, b)\).

Step 1: Understanding Recitations of Concavity

When the second derivative \(f''(x) < 0\), it indicates that the function \(f(x)\) is concave down on the interval \((a, b)\). This means the function is shaped like an upside-down bowl over this interval.

Step 2: Analyzing Critical Points

The critical points of a function \(f(x)\) occur where the first derivative \(f'(x) = 0\) or where \(f'(x)\) is undefined. For a concave down function, if \(f'(x) = 0\) occurs, it usually happens at the unique maximum within the interval.

Step 3: Applying the Concavity Test

Given the function is concave down on \((a, b)\), if \(f'(x)\) becomes zero, it only does so once, since multiple zero crossings would indicate a change in concavity which contradicts \(f''(x) < 0\). Therefore, \(f'(x)\) can be zero at most once.

Step 4: Concluding the Solution

Since \(f'(x)\) is zero at most once in \((a, b)\), the correct answer is option (B): "at most once in \((a, b)\)."

Key Concepts

Understanding the Second Derivative

The second derivative, represented as \(f''(x)\), is a powerful tool in calculus that provides valuable insight into the behavior of a function. It is the derivative of the first derivative and tells us about the concavity of the function.

• When \(f''(x) > 0\), the function is concave up, resembling a U-shape.
• Conversely, when \(f''(x) < 0\), the function is concave down, resembling an upside-down U or a bowl turned over.

These insights are essential because they help identify the nature of the function's curve over an interval, giving us a clear picture of whether the function has a peak or valley within a given range.

Critical Points and Their Significance

Critical points of a function occur where the first derivative, \(f'(x)\), is zero or undefined. These are the points where the function potentially has a local minimum, maximum, or a point of inflection.

• If \(f'(x) = 0\), this may indicate a horizontal tangent line, suggesting a possible maximum or minimum.
• If \(f'(x)\) is undefined, this might occur at a sharp corner or cusp in the function's graph.

Understanding critical points is crucial as they are candidates where changes in the function occur, and thus, these points help in determining the function's overall behavior and key characteristics.

Exploring the First Derivative

The first derivative, denoted as \(f'(x)\), is fundamental in understanding a function's rate of change. It provides us information about how the function is increasing or decreasing.

• When \(f'(x) > 0\), the function is increasing; the graph goes upward as we move along the x-axis.
• When \(f'(x) < 0\), the function is decreasing; the graph goes downward.
• Where \(f'(x) = 0\), we may find critical points which could indicate peaks (maximas) or troughs (minimas) in the function.

By analyzing how \(f'(x)\) behaves, we can predict the patterns and trends of the function, which is invaluable for deeper function analysis.

Interval Analysis and Its Importance

Interval analysis involves examining the behavior of a function over specific parts of its domain. By segmenting a function into intervals, we can more easily describe where the function increases or decreases, and whether it is concave up or down.

• By identifying where the first or second derivatives change signs, we establish intervals over which the function has different characteristics.
• Important intervals arise from breaking points identified by the first derivative being zero or the second derivative changing signs.

Effective interval analysis is crucial for solving complex calculus problems, such as finding where the function reaches maximum or minimum values or understanding the curvature and shape of the graph overall.