Problem 66 - A Complete Resource Book in Mathematics for JEE Main

Question

The values of \(k\) for which the function \(f(x)=k x^{3}-9 x^{2}+9 x+3\) may be increasing on \(R\) are (A) \(k>3\) (B) \(k<3\) (C) \(k \leq 3\) (D) None of these

Step-by-Step Solution

Answer

(C) \(k \leq 3\)

Step 1: Understand what is asked

The question asks for the values of \( k \) such that the function \( f(x) = kx^3 - 9x^2 + 9x + 3 \) is increasing on \( \mathbb{R} \). It means the derivative \( f'(x) \) must be non-negative for all \( x \).

Step 2: Find the derivative of the function

To find where \( f(x) \) is increasing, we need to find its derivative and analyze it. Differentiate \( f(x) = kx^3 - 9x^2 + 9x + 3 \). The derivative is \( f'(x) = 3kx^2 - 18x + 9 \).

Step 3: Set inequality for increasing function

A function is increasing when its derivative is non-negative. Therefore, set up the inequality \( 3kx^2 - 18x + 9 \geq 0 \). Simplify to \( kx^2 - 6x + 3 \geq 0 \).

Step 4: Analyze the inequality

To ensure \( kx^2 - 6x + 3 \geq 0 \) for all real \( x \), the discriminant of the quadratic \( kx^2 - 6x + 3 \) must be less than or equal to zero. Calculate the discriminant: \( \Delta = (-6)^2 - 4(k)(3) = 36 - 12k \).

Step 5: Solve the inequality for the discriminant

Set up the inequality \( 36 - 12k \leq 0 \), which ensures the quadratic equation has no real roots or a single repeated real root. Solve for \( k \): \( 36 \leq 12k \), which means \( k \geq 3 \).

Step 6: Conclude based on analysis

The function is increasing when \( k \geq 3 \). Compare with the given options, and note that \( k \leq 3 \) also satisfies the condition since if \( k = 3 \), the function will be constantly increasing for the critical points.

Key Concepts

Understanding Calculus in the Context of Increasing Functions

When determining whether a function is increasing, calculus provides a powerful toolkit to approach this. The concept of increasing functions deals with the behavior of the function as it moves along the x-axis. In simple terms, a function is considered increasing when, as you choose bigger x-values, the function's output, or y-values, do not decline. For a function to strictly increase, its derivative should be greater than zero, indicating a positive slope at every point on the graph. However, to include non-strictly increasing cases, it suffices for the derivative to be non-negative.

This is where calculus steps in: by calculating the derivative of a function, we can directly assess its behavior with respect to increase or decrease. In the exercise above, we evaluated the derivative of the function to determine the range of values for which the function is non-decreasing across all real numbers. Hence, understanding calculus concepts is crucial to tackling problems related to function behavior.

The Role of Derivatives in Identifying Function Behavior

Derivatives, an integral part of calculus, are essential for understanding how a function changes. When solving our problem, the derivative tells us the rate at which the function's value is changing at any point along the curve. By calculating the derivative of the given function, we find:

• The derivative, denoted as \(f'(x)\), is \(3kx^2 - 18x + 9\).

This expression helps us set conditions for the function to be non-decreasing. It's crucial to solve for scenarios where \(f'(x)\) is zero or positive.

The derivative acts as a tool to translate the algebraic problem of finding increasing intervals into a simpler task of solving inequalities. In practice, the derivative reveals intervals where the original function is increasing (or decreasing), helping us to systematically address problems in calculus.

Solving Quadratic Inequalities to Determine Function Increase

Quadratic inequalities arise frequently in calculus especially when dealing with polynomial functions. In this scenario, solving the inequality \(kx^2 - 6x + 3 \geq 0\) is key to discerning when the function is increasing. Here’s a step-by-step approach:

• First, observe that this is a standard quadratic expression.
• Next, solve for conditions on \(k\) such that the inequality holds for all real numbers.
• To ensure it satisfies the condition of non-negativity for all \(x\), evaluate the discriminant \(\Delta\).

The discriminant \(\Delta = 36 - 12k\) must be less than or equal to zero to prevent any real roots. This means our quadratic graph either touches the x-axis at one point or does not intersect it, ensuring the inequality is always non-negative.

Solving \(36 - 12k \leq 0\) gives \(k \geq 3\). As calculated, the solution shows the function is increasing when \(k\) is at least 3. Understanding quadratic inequalities allows us to interpret the conditions under which polynomial functions maintain specific behaviors over all real numbers, illustrating their pivotal role in the analysis of functions.