Problem 52 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(f(1)=-2\) and \(f^{\prime}(x) \geq 4.2\) for \(1 \leq x \leq 6 .\) The possible value of \(f(6)\) lies in the interval : \(\quad\) [April 25, 2013] (a) \([15,19)\) (b) \((-\infty, 12)\) (c) \([12,15)\) (d) \([19, \infty)\)

Step-by-Step Solution

Answer

The value of \( f(6) \) is at least 19, so it lies in the interval \([19, \infty)\).

Step 1: Understanding the Problem

We are given that \( f(1) = -2 \) and the derivative \( f'(x) \) is at least 4.2 for \( 1 \leq x \leq 6 \). We need to determine the range for \( f(6) \).

Step 2: Estimating the Change in f(x)

The derivative \( f'(x) \) being at least 4.2 implies that \( f(x) \) is increasing at a rate of at least 4.2 per unit increase in \( x \). This means the function value increases by at least \((4.2) \times (6-1) = 21\) over the interval \(x = 1\) to \(x = 6\).

Step 3: Calculating f(x) for x=6

Given \( f(1) = -2 \), the increase of at least 21 units means \( f(6) \geq -2 + 21 = 19 \). Therefore, the minimum possible value of \( f(6) \) is 19.

Step 4: Defining the Interval

Since \( f(6) \geq 19 \), the function value at \( x=6 \) lies in the interval \( [19, \infty) \). Thus, the correct answer is option (d) \([19, \infty)\).

Key Concepts

Exploring Calculus

Calculus is a branch of mathematics that focuses on the study of change. It provides the tools necessary to analyze the dynamic lives of functions and how they behave along different intervals of the real line. The Mean Value Theorem (MVT) is a fundamental principle in calculus. This theorem tells us about the "average" behavior of a function over an interval. By using derivatives, calculus allows us to predict how functions will behave by examining local linear approximations, which helps us track the rate at which function values rise or fall.

• Calculus helps us understand how quickly quantities increase or decrease.
• MVT gives us insights into the overall progression of a function by considering its derivatives.
• Through derivatives, calculus guides us in estimating instantaneous rates of change.

By applying these concepts, students can better analyze abstract problems like the one given, making them more tangible and understandable.

Function Behavior Insight

Functions can describe relationships between two sets of numbers or quantities, and understanding their behavior can provide valuable insights into real-world phenomena. The original problem involves us figuring out how a function behaves over a specific interval. We know the function starts at a value of -2, and since the derivative is always greater than or equal to 4.2, we understand that the function consistently increases at least this fast.

• A function's behavior is guided by its derivative, which indicates direction and rate of change.
• In this exercise, the derivative informs us that the function increases from 1 to 6.
• This consistent increase over the range dictates that the function value for \( f(6) \) must be in the range \( [19, \infty) \).

This helps us conclude that the function is always on an upward trajectory throughout the specified interval, ultimately reaching values of at least 19 at its endpoint.

Estimating Derivatives

Estimating the behavior of a function often requires us to understand how its derivatives work. In our exercise, we know the derivative, \( f'(x) \), is at least 4.2 over the interval from 1 to 6. Essentially, this derivative tells us how steeply the function increases. The derivative represents the slope of the tangent line to the curve, effectively showing the function's rate of change.

• The derivative helps in understanding the local behavior of the function at any point.
• An estimate from the derivative provides minimum and maximum bounds for how much the function value changes over a given interval.
• In our example, since the derivative is \( \geq 4.2 \), the function increases by at least \( 4.2 \times (6 - 1) = 21 \) over the interval.

Ultimately, estimating with derivatives offers a powerful way to understand function progress without knowing its exact equation, a concept crucial in various scientific and engineering applications.