Question

If \(\mathrm{f}(\mathrm{x})\) is a differentiable function satisfying \(f^{\prime}(x)<2\) for all \(x \in R\) and \(f(1)=2\), then greatest possible integral value of \(f(3)\) is (1) 5 (2) 6 (3) 4 (4) Data insufficient

Short answer

5

Step-by-step solution

Understand the Given Information

The function \(\f(x)\) is differentiable and its derivative \(f'(x) < 2 \) for all \( x \) in the real numbers. Also, it is given that \(\f(1) = 2 \).

Interpret the Derivative Condition

Since \(f'(x) < 2 \), it means the slope of the function \(\f(x)\) is always less than 2 for any value of \( x \).

Set Up the Integral Inequality

We need to find \(f(3) \). Start by using the fact that the slope of the secant line from x = 1 to x = 3 should be less than 2: \[ \frac{f(3) - f(1)}{3 - 1} < 2 \].

Simplify the Inequality

Substitute \( f(1) = 2 \) into the inequality: \[ \frac{f(3) - 2}{2} < 2 \].

Solve for \( f(3) \)

Multiply both sides by 2: \[ f(3) - 2 < 4 \]. Then, add 2 to both sides: \[ f(3) < 6 \].

Determine the Greatest Integral Value

The greatest integral value less than 6 is 5.

Key concepts

Differentiable Functions

A function is said to be differentiable at a point if it has a derivative at that point. This means that the function's graph has a tangent line that touches it smoothly at that specific location. For a function to be differentiable over an interval, it must be smooth and not have any sharp corners or discontinuities within that interval. Imagine drawing the graph without lifting your pen—you should be able to do this for a differentiable function.

Differentiable functions are important because they guarantee that small changes in the input result in small changes in the output. This property is useful in various fields like physics and engineering, where continuous and predictable behavior is critical. In our exercise, we are given a differentiable function \(\f(x)\). This assures us that the function behaves smoothly across the entire real line.

Derivatives

The derivative of a function measures how fast its value changes as its input changes. In other words, it tells us the slope of the function at any given point. The concept of the derivative is central to calculus and has numerous applications, from calculating velocity to optimizing complex systems.

In the given problem, we know that \(\f'(x) < 2\) for all \(\f(x)\). This means that the slope of the function at any point is always less than 2. Imagine this as a constraint that limits how steep the function's graph can be. No matter where you are on the graph, it can never rise faster than a slope of 2. This information is critical in setting up the inequality to find \(\f(3)\). By knowing the upper limit of the derivative, we can predict how the function behaves over an interval.

Integral Inequalities

Integral inequalities help us understand relationships between functions over intervals. They give us bounds on how much a function can grow or decrease over an interval, which is particularly useful in optimization problems.

In the exercise, we use an inequality involving the integral of the derivative to find the greatest possible value of \(\f(3)\). Here’s how:

• We start with the inequality derived from the derivative constraint: \(\frac{f(3) - f(1)}{3 - 1} < 2\).
• By substituting \(\f(1) = 2\), we get \(\frac{f(3) - 2}{2} < 2\).
• Simplify this to get \(\f(3) < 6\), implying that the highest value \(\f(3)\) can achieve without violating the derivative constraint is just under 6.

This method of leveraging integral inequalities and derivative constraints allows us to bound the solution, ensuring we stay within the limits imposed by the given conditions. The final step is to determine the greatest possible integral value under these constraints, which is 5.