Problem 114 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(f\) be differentiable for all \(x\). If \(f(1)=-2\) and \(f^{\prime}(x) \geq 2\) for \(x \in[1,6]\), then (a) \(f(6) \geq 8(\) b \() f(6)<8\) (c) \(f(6)<5\) (d) \(f(6)=5\)

Step-by-Step Solution

Answer

The correct answer is (a) \( f(6) \geq 8 \).

Step 1: Understand the Problem

The function \( f \) is differentiable for all \( x \), which means it is continuous and well-behaved on the interval \([1, 6]\). We are given initial condition \( f(1) = -2 \) and that the derivative \( f'(x) \) is at least 2 for all \( x \) in \([1, 6]\). Our task is to determine the possible value of \( f(6) \).

Step 2: Apply the Mean Value Theorem

By the Mean Value Theorem, if \( f \) is continuous on \([1, 6]\) and differentiable on \((1, 6)\), then there exists some \( c \) in \((1, 6)\) such that \( f'(c) = \frac{f(6) - f(1)}{6 - 1} \). Since \( f'(x) \geq 2 \) for all \( x \) in this interval, we know that \( \frac{f(6) - (-2)}{5} \geq 2 \).

Step 3: Solve the Inequality

Solve the inequality \( \frac{f(6) + 2}{5} \geq 2 \):1. Multiply both sides by 5: \[ f(6) + 2 \geq 10 \]2. Subtract 2 from both sides: \[ f(6) \geq 8 \]

Step 4: Determine the Correct Option

From the solved inequality \( f(6) \geq 8 \), we can conclude that the correct answer is part (a). We know this condition aligns with the given derivative constraint over the interval.

Key Concepts

Differentiable Functions

When we talk about a differentiable function, we are referring to a function that has a derivative everywhere within its domain. This means that the function shows a smooth and predictable change without any sharp corners or breaks.

A differentiable function is automatically continuous. However, the reverse is not necessarily true; not all continuous functions are differentiable. For example, the absolute value function is continuous but not differentiable at zero due to its sharp corner.

In the given exercise, function \( f \) is differentiable on the interval \([1, 6]\), ensuring it doesn't have any abrupt changes within this region. This information tells us that \( f \) maintains a smooth curve throughout the interval, allowing us to apply calculus concepts like the Mean Value Theorem.

Derivative Inequality

The derivative of a function tells us how the function's output value changes with respect to changes in the input value. It gives us a rate of change at any particular point on a graph.

In the exercise, it's provided that \( f'(x) \geq 2 \) for all \( x \) in the interval \([1, 6]\). This is a specific form of what we call a derivative inequality. Such inequalities set boundaries or conditions for the rate of change.

Here, the derivative inequality states that the rate of change of \( f \) is never less than 2. It implies that as \( x \) increases from 1 to 6, the function is, at the very least, increasing steadily rather than decreasing or slowing down. This condition is crucial when applying the Mean Value Theorem to estimate or determine the actual values that \( f \) could take at endpoints.

Continuous Functions

Continuous functions are those which do not have any abrupt interruptions or jumps in their graphs. You can draw continuous functions without lifting your pencil from the paper, ensuring smooth connectivity between points.

For a function to be continuous on an interval, it must be defined at every point in that interval, and its limit at any point within the interval must equal the function's value at that point.

This property is imperative in applying the Mean Value Theorem. In the problem, \( f \) is declared continuous over \([1, 6]\). This continuity, along with differentiability, allows us to apply such theorems effectively. Without continuity, results from theorems like the Mean Value Theorem would not hold, as they require uninterrupted behavior of the function over the interval.