Problem 166 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f\) be differentiable for all \(x\). If \(f(1)=-2\) and \(f^{\prime}(x) \geq 2\) for \(x \in[1,6]\), then \(\quad\) [2005] (A) \(f(6) \geq 8\) (B) \(f(6)<8\) (C) \(f(6)<5\) (D) \(f(6)=5\)

Step-by-Step Solution

Answer

(A) \( f(6) \geq 8 \) is correct.

Step 1: Understand the Problem

We are given that the function \( f \) is differentiable for all \( x \). It is also known that \( f(1) = -2 \) and the derivative \( f'(x) \geq 2 \) for \( x \in [1,6] \). We need to determine the correct statement about \( f(6) \).

Step 2: Apply the Mean Value Theorem

The Mean Value Theorem states that for a differentiable function \( f \) on the interval \([a,b]\), there exists \( c \in (a,b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b-a} \). Here, we apply it on the interval \([1,6]\).

Step 3: Set Up the Inequality

Using the Mean Value Theorem, we have:\[f'(c) = \frac{f(6) - f(1)}{6-1} = \frac{f(6) + 2}{5}.\]Since it is given that \( f'(x) \geq 2 \), we have\[\frac{f(6) + 2}{5} \geq 2.\]

Step 4: Solve the Inequality

Multiply both sides by 5 to clear the fraction:\[f(6) + 2 \geq 10.\]Subtract 2 from both sides to solve for \( f(6) \):\[f(6) \geq 8.\]

Step 5: Conclusion

Based on our solution, \( f(6) \geq 8 \). Therefore, the correct answer is option (A): \( f(6) \geq 8 \).

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a key result in calculus, particularly when discussing differentiability and the behavior of functions over an interval. This theorem essentially says that for any given section of a continuous and differentiable curve, there is at least one point where the instantaneous rate of change (the derivative) matches the average rate of change over that interval. This important concept is often encapsulated in the formula:

• If a function \( f \) is continuous over \([a, b]\) and differentiable over \((a, b)\), then there persists a point \( c \) in \((a, b)\) such that \( f'(c) = \frac{f(b) - f(a)}{b-a} \).

When applying this to problems, like in the original exercise, ensure the function is both continuous and differentiable over the given interval. This will validate the existence of the point \( c \). The theorem simplifies to finding that somewhere on the curve between two points, the slope of the tangent is equal to the slope of the secant line joining those points. This is an immensely useful tool in calculus because it bridges the gap between average and instantaneous rates of change.

Inequality

Inequalities are expressions that show the relative size or order of two values. When dealing with inequalities in calculus, they often express constraints or conditions that a function or its derivative must satisfy. In the given exercise, the statement \( f'(x) \geq 2 \) for \( x \in [1,6] \) represents an inequality constraint on the derivative of the function.

• An inequality like \( f'(x) \geq 2 \) suggests that the rate of change of the function is never less than 2 within the interval.
• This can affect the behavior of the function dramatically, ensuring that \( f(x) \) is increasing at a rate of at least 2 units per unit increase in \( x \).

Solving inequalities involves finding the set of all possible values that make the inequality true. In the exercise, this helps us deduce that \( f(6) \) has a minimum value based on the initial condition \( f(1) = -2 \), leading logically to the conclusion that \( f(6) \geq 8 \). In calculus, handling inequalities often involves algebraic manipulation, such as multiplying or dividing the entire inequality by constants, while keeping in mind the rules of inequality operations.

Derivative concept

Understanding derivatives is central to calculus and mathematical analysis. The derivative represents the rate at which a quantity changes. It's often described as the slope of the tangent line at any point on a graph.

• A derivative, denoted as \( f'(x) \), provides us with a way to determine the behavior of the function—whether it's increasing, decreasing, or constant.
• In the context of the exercise, \( f'(x) \geq 2 \) implies a steady or accelerating increase in the function within the interval \([1,6]\).

Knowing how to evaluate and interpret derivatives can help **predict function behavior**, make both qualitative and quantitative predictions about how functions vary, and solve real-world problems involving rates of change. For instance, if \( f'(x) \) holds a constant minimum rate of change as given, not only does it guide us in solving inequalities, but also helps justify the Mean Value Theorem application. This highlights derivatives as both a mathematical tool for analysis and a concept for proving broader mathematical truths.