Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The range of values of \(a\) for which the function \(f(x)=x^{3}+(a+2) x^{2}+3 a x+5\) may be monotonic in \(R\), is (A) \(a<1\) (B) \(14\) (D) None of these

Step-by-Step Solution

Answer

The function is monotonic for \( 1 < a < 4 \). Option (B) is correct.

Step 1: Understanding Monotonicity

A function is monotonic on an interval if it is either entirely non-increasing or non-decreasing throughout that interval. To determine when a function is monotonic on the entire real line, we check the derivative.

Step 2: Differentiate the Function

Differentiating the function, we find:\[ f'(x) = \frac{d}{dx} \, \left( x^{3} + (a+2) x^{2} + 3a x + 5 \right) = 3x^{2} + 2(a+2)x + 3a. \]

Step 3: Analyze the Derivative

To ensure the function is monotonic on \( \mathbb{R} \), the derivative \( f'(x) \) should not change sign. Therefore, it should either be always non-negative or non-positive.

Step 4: Determine Sign of Quadratic Derivative

The derivative \( f'(x) = 3x^{2} + 2(a+2)x + 3a \) is a quadratic expression. Its sign over the entire real line depends on its discriminant \( \Delta = b^{2} - 4ac \). Here, \( a = 3, b = 2(a+2), c = 3a \).

Step 5: Calculate the Discriminant

Calculate the discriminant of the quadratic:\[\Delta = \left( 2(a+2) \right)^{2} - 4 \times 3 \times 3a \]\[\Delta = 4(a+2)^2 - 36a \]

Step 6: Simplify the Discriminant

Simplify the expression of the discriminant:\[\Delta = 4(a^2 + 4a + 4) - 36a \]\[= 4a^2 + 16a + 16 - 36a \]\[= 4a^2 - 20a + 16 \]\[= 4(a^2 - 5a + 4)\]

Step 7: Solve for Discriminant Condition

For the function to be monotonic, the quadratic should be a perfect square, i.e., \( \Delta \leq 0 \). Solve for \( a \, \) such that \( 4(a^2 - 5a + 4) \leq 0 \).

Step 8: Solve \( a^2 - 5a + 4 \leq 0 \)

Consider the roots of the quadratic \( a^2 - 5a + 4 = 0 \). Solving:\[a = \frac{5 \pm \sqrt{25-16}}{2} = \frac{5 \pm 3}{2}\]\[a = 4 \quad \text{or} \quad a = 1 \]The inequality \( a^2 - 5a + 4 \leq 0 \) is satisfied for \( 1 \leq a \leq 4 \).

Step 9: Conclusion on Monotonicity

Thus, for the function \( f(x) \) to be monotonic over \( \mathbb{R} \), \( a \) must satisfy \( 1 \leq a \leq 4 \). Therefore, the range of values for \( a \) is option (B): \( 1 < a < 4 \).

Key Concepts

Derivatives

Derivatives are fundamental in understanding how functions behave. They provide us with the rate of change of the function with respect to its variables. In simpler terms, a derivative tells us how fast or slow a function's output is changing as the input changes.

For a function to be monotonic, its derivative must have a consistent sign over an interval. It should either remain positive or negative throughout the interval. This means that the function is either consistently increasing or decreasing, respectively.

• Consistent positive derivative: Function is increasing.
• Consistent negative derivative: Function is decreasing.

Analyzing derivatives helps in understanding the critical points and the behavior of curves over the real line. For the function in question, the derivative was calculated as: \[ f'(x) = 3x^{2} + 2(a+2)x + 3a. \]This derivative is a quadratic expression that we will need to study further to determine its behavior.

Quadratic Expressions

Quadratic expressions are polynomials of degree 2 and have the standard form \( ax^2 + bx + c \). In many scenarios, such as determining monotonicity, they are powerful tools.

Quadratic expressions can either be:

• Positive definite: Always positive regardless of the input, indicating that the original function is always increasing.
• Negative definite: Always negative, indicating the function is always decreasing.
• Indefinite: Changes sign, which means the function is neither strictly increasing nor decreasing.

In our problem, the expression for the derivative is quadratic:\[ 3x^{2} + 2(a+2)x + 3a \]Once we have the quadratic expression, the next step is to analyze its discriminant to understand its sign behavior.

Discriminant

The discriminant is a key concept in the study of quadratic expressions. Given a quadratic expression \( ax^2 + bx + c \), the discriminant \( \Delta \) is calculated as:\[ \Delta = b^2 - 4ac \]The value of \( \Delta \) provides crucial information about the roots and the nature of the quadratic:

• If \( \Delta > 0 \), the quadratic has two distinct real roots and can change signs, meaning the function derived may not be monotonic.
• If \( \Delta = 0 \), the quadratic has a repeated real root, creating a perfect square, indicating the derivative cannot change sign.
• If \( \Delta < 0 \), there are no real roots, the quadratic does not change sign, and the function remains consistently increasing or decreasing.

In the problem context, the discriminant is:\[ \Delta = 4(a^2 - 5a + 4) \]The condition \( \Delta \leq 0 \) implies the derivative remains the same sign, ensuring monotonicity across \( \mathbb{R} \). Solving this returns the range of values for \( a \), which ensures the function remains monotonic.