Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The least possible value of \(k\) for which the function \(f(x)=x^{2}+k x+1\) may be increasing on \([1,2]\) is (A) 2 (B) \(-2\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The least possible value of \(k\) is \(-2\). Option B.

Step 1: Understand the conditions for an increasing function

A function is increasing on an interval if its derivative is non-negative for every point within that interval. For the function given, \( f(x) = x^2 + kx + 1 \), we need to find the derivative and check the conditions under which this derivative is non-negative on \([1, 2]\).

Step 2: Differentiate the function

Find the derivative of \( f(x) \) with respect to \( x \). The derivative, \( f'(x) \), is given by:\[ f'(x) = 2x + k \]

Step 3: Set up the inequality for increasing function

The function is increasing on \([1, 2]\) if \( f'(x) = 2x + k \geq 0 \) for all \( x \) in the interval \([1, 2]\). This implies two inequalities: 1. \( 2(1) + k \geq 0 \) 2. \( 2(2) + k \geq 0 \)

Step 4: Solve the inequalities

Solve the inequalities derived in the previous step:- From \( 2(1) + k \geq 0 \), we get \( 2 + k \geq 0 \) or \( k \geq -2 \).- From \( 2(2) + k \geq 0 \), we get \( 4 + k \geq 0 \) or \( k \geq -4 \).The stronger condition is \( k \geq -2 \), as it encompasses \( k \geq -4 \).

Step 5: Determine the least possible value of \( k \)

The least possible value of \( k \) that satisfies both conditions is \( k = -2 \). Thus, \( k = -2 \) is the smallest value for which the function is increasing on \([1, 2]\).

Key Concepts

Increasing Function

When we talk about an increasing function, it means that as you move along the graph from left to right, the values of the function never decrease.
An increasing function keeps climbing, or at least stays constant, over a specified interval.

If we consider a function \( f(x) \), it is increasing on a closed interval \([a, b]\) when for any two numbers \( x_1 \) and \( x_2 \) in that interval where \( x_1 < x_2 \), the function values satisfy:

• \( f(x_1) \leq f(x_2) \).

This simply means that the output doesn't drop as the input rises. For differentiable functions, like many of those encountered in calculus, this concept links closely to their derivatives. That brings us to our next topic, derivatives, which helps in looking at the rate of change of these functions.

Derivative

The derivative of a function provides a powerful tool in calculus, representing the rate of change of the function's output for a given change in input.
The derivative essentially gives us the slope of the tangent line at any point on the function's graph.

For a function \( f(x) \), the derivative, denoted as \( f'(x) \), captures how the function’s values change as \( x \) changes.

• If \( f'(x) > 0 \), the function is increasing at that point.
• If \( f'(x) = 0 \), the function is constant (flat) at that point.
• If \( f'(x) < 0 \), the function is decreasing at that point.

This knowledge lets us examine whether a function increases or decreases across intervals by checking the sign of the derivative throughout that range. Let's see how this approach applies practically when solving inequalities.

Inequality Solving

Solving inequalities is a key part of determining where a function is increasing or decreasing.
The inequalities arise naturally from the derivative's conditions: checking whether it is non-negative or negative over certain intervals.

Consider the derivative we found: \( f'(x) = 2x + k \). To identify where the function \( f(x) = x^2 + kx + 1 \) is increasing, we set up the inequality:

• \( 2x + k \geq 0 \).

This inequality must hold true for all \( x \) in the interval \([1, 2]\). To solve, we plug the interval endpoints into our inequality:

• At \( x = 1 \): \( 2(1) + k \geq 0 \), resulting in \( k \geq -2 \).
• At \( x = 2 \): \( 2(2) + k \geq 0 \), giving \( k \geq -4 \).

Comparing the results tells us the strongest condition, which is \( k \geq -2 \).
Through solving such inequalities, you can identify conditions like the smallest value of \( k \) for which a function remains increasing over a given interval.