Problem 14 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(f(x)=k x^{3}-9 x^{2}+9 x+3\) is monotonically increasing in each interval, then (a) \(k<3\) (b) \(k \leq 3\) (c) \(k>3\) (d) None of these

Step-by-Step Solution

Answer

The correct option is (c) \( k>3 \).

Step 1: Understanding Monotonic Function

For a function to be monotonically increasing, its derivative must be non-negative for all values in its domain. This means we need to find the derivative and set conditions for it to be greater than or equal to zero.

Step 2: Find the Derivative

Derive the function \( f(x) = kx^3 - 9x^2 + 9x + 3 \). The derivative is \( f'(x) = 3kx^2 - 18x + 9 \).

Step 3: Set the Derivative as Non-negative

To ensure the function is monotonically increasing, set \( f'(x) \geq 0 \). Therefore, \( 3kx^2 - 18x + 9 \geq 0 \).

Step 4: Simplify the Inequality

Simplify the inequality \( 3kx^2 - 18x + 9 \geq 0 \) by dividing the entire inequality by 3: \( kx^2 - 6x + 3 \geq 0 \).

Step 5: Analyzing the Quadratic Inequality

For the quadratic \( kx^2 - 6x + 3 \geq 0 \) to be positive for all \( x \), the determinant (discriminant) must be less than or equal to zero.

Step 6: Calculating the Discriminant

The discriminant of the quadratic equation \( kx^2 - 6x + 3 \) is \( (-6)^2 - 4(k)(3) \), leading to \( 36 - 12k \).

Step 7: Condition for Non-positive Discriminant

To ensure the quadratic is non-positive, set \( 36 - 12k \leq 0 \).

Step 8: Solve the Discriminant Inequality

Solving \( 36 - 12k \leq 0 \) provides \( k \geq 3 \).

Step 9: Interpret the Solution

The solution \( k \geq 3 \) means the function can never be monotonically increasing for k less than 3. Hence, only \( k > 3 \) makes hold for all \( x \).

Step 10: Conclusion

Therefore, for the function to be monotonically increasing in its intervals, the correct condition for \( k \) is \( k > 3 \).

Key Concepts

Derivative of a function

The derivative of a function is a measure of how the function's output changes with respect to a change in its input. This is a fundamental concept in calculus and is essential for understanding the behavior of functions.
The derivative can be viewed as the slope of the tangent line to the graph of the function. It gives us the rate at which the function is increasing or decreasing at any given point.
For a function like \( f(x) = kx^3 - 9x^2 + 9x + 3 \), calculating the derivative involves applying standard derivative rules.

• The power rule says that the derivative of \( x^n \) is \( nx^{n-1} \).
• Combine like terms accordingly, which results in \( f'(x) = 3kx^2 - 18x + 9 \).

Understanding this derivative allows us to analyze how \( f(x) \) behaves in terms of increasing and decreasing, which brings us to the topic of monotonic functions.

Quadratic discriminant

The discriminant is a part of the quadratic formula and plays a critical role in determining the nature of the roots of a quadratic equation. If the quadratic equation is written in the form \( ax^2 + bx + c \), the discriminant is given by \( b^2 - 4ac \).
The discriminant helps us understand:

• If the discriminant is positive, the quadratic has two distinct real roots.
• If it is zero, the quadratic has exactly one real root (a repeated root).
• If negative, the quadratic has no real roots (the roots are complex).

In the context of our exercise, the quadratic inequality \( kx^2 - 6x + 3 \geq 0 \) requires the discriminant \( 36 - 12k \) to be less than or equal to zero. This ensures that the quadratic expression does not change sign around any real roots, thus guaranteeing it remains non-negative for all \( x \). Thus, solving \( 36 - 12k \leq 0 \) will guide us to the correct values of \( k \) that make the function monotonically increasing.

Inequality solving

Inequality solving involves finding ranges of values for which expressions satisfy a certain condition. In our problem, we worked with the inequality derived from the derivative of the function.
Upon deriving the inequality \( kx^2 - 6x + 3 \geq 0 \), we simplify this by first analyzing its discriminant.
Solving \( 36 - 12k \leq 0 \) gives:

• Rearrange to find \( 36 \leq 12k \).
• Divide by 12 to isolate \( k \), resulting in \( k \geq 3 \).

However, for monotonic increase, \( k \) should strictly be greater than 3 to ensure that the derivative inequality holds for all x without exception.
This technique highlights the process of solving quadratic inequalities by leveraging their components and understanding their behavior.