Problem 67 - 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 \(-2\).

Step 1: Determine the derivative

To find where the function \(f(x) = x^2 + kx + 1\) is increasing, we first determine its derivative. The derivative is given by \(f'(x) = 2x + k\).

Step 2: Set conditions for increasing function

A function is increasing on an interval if its derivative is non-negative on that interval. We need \(f'(x) \geq 0\) for all \(x\) in the interval \([1,2]\).

Step 3: Apply the interval endpoints

Substitute the endpoint values into the derivative: - At \(x = 1\), we have \(f'(1) = 2 \cdot 1 + k = 2 + k\).- At \(x = 2\), we have \(f'(2) = 2 \cdot 2 + k = 4 + k\).

Step 4: Solve inequalities for increasing condition

We require that both inequalities hold: 1. \(2 + k \geq 0\), which simplifies to \(k \geq -2\).2. \(4 + k \geq 0\), which simplifies to \(k \geq -4\).

Step 5: Determine least possible value of \(k\) for increasing behavior

Since \(k\) must satisfy both \(k \geq -2\) and \(k \geq -4\), the least possible value satisfying both is \(k = -2\).

Key Concepts

Derivatives

When solving mathematics problems involving functions, understanding derivatives is crucial. A derivative, in simple terms, describes how a function changes. It is a measure of how a function's output value changes as its input changes. For example, the derivative of a position with respect to time is velocity, indicating how fast the position changes over time.

In mathematical notation, the derivative of a function \( f(x) \), represented as \( f'(x) \), informs us about the slope of the tangent line at any point on the function's curve. If \( f'(x) \) is positive at a point, the function is increasing there. If it's negative, the function is decreasing.

For the function \( f(x) = x^2 + kx + 1 \), the derivative is \( f'(x) = 2x + k \). This derivative shows us how changes in \( x \) influence \( f(x) \) depending on the value of \( k \). By setting up condition \( f'(x) \geq 0 \), we're identifying when and where the function increases.

Increasing Functions

An increasing function is one where as the input (\( x \)) increases, the function's output does not decrease. This means that as you move along the graph from left to right, you either move upwards or stay horizontal.

A function is strictly increasing if \( f'(x) > 0 \) for all values within a given interval. On the other hand, it is non-decreasing or weakly increasing if \( f'(x) \geq 0 \), which allows for flat regions where the function's value doesn't change.

In our problem, we examined \( f'(x) = 2x + k \) to ensure it remains non-negative over the interval \([1, 2]\). By checking endpoints \( x = 1 \) and \( x = 2 \), and ensuring \( f'(x) \geq 0 \) there, we determine the smallest \( k \) to maintain the function's non-decreasing nature over \([1, 2]\).

Inequalities

Inequalities are mathematical expressions involving the symbols \(<\), \(>\), \( \leq \), and \( \geq \). They help define ranges of possible values for variables where the expressions hold true. Inequalities are extensively used in calculus to express conditions like a derivative being non-negative or non-positive across an interval.

In our function problem, inequalities are crucial to determine the appropriate values of \( k \) so that the function is increasing. We specifically solve inequalities:

• \( 2 + k \geq 0 \), simplifying to \( k \geq -2 \)
• \( 4 + k \geq 0 \), simplifying to \( k \geq -4 \)

These conditions must be satisfied simultaneously, leading us to choose the highest lower bound of \( k \), which is \(-2\). This ensures the derivative, and therefore the function, is non-negative over the entire interval \([1, 2]\), identifying \( k = -2 \) as the smallest value that achieves this.