Problem 165 - A Complete Resource Book in Mathematics for JEE Main

Question

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? Interval Function (A) \((-\infty, \infty)\) \(x^{3}-3 x^{2}+3 x+3\) (B) \([2, \infty)\) \(2 x^{3} 3 x^{2}-12 x+6\) (C) \(\left(-\infty, \frac{1}{3}\right]\) \(3 x^{2}-2 x+1\) (D) \((-\infty,-4]\) \(x^{3}+6 x^{2}+6\)

Step-by-Step Solution

Answer

Pair (C) is incorrectly matched.

Step 1: Determine the Critical Points

For each function, find its derivative and set it equal to zero to determine the critical points where the function's behavior may change.

Step 2: Analyze Function (A)

The derivative of \(x^{3}-3x^{2}+3x+3\) is \(3x^{2}-6x+3\). Setting this equal to zero gives \(3x^{2}-6x+3=0\). Solving, we find the critical points are \(x=1\). Thus, the function is increasing for \(x<1\) and \(x>1\) due to the positive lead term.

Step 3: Analyze Function (B)

The derivative of \(2x^{3}+3x^{2}-12x+6\) is \(6x^{2}+6x-12\). Setting this equal to zero gives \(6(x^2+x-2)=0\), factoring to \((x-1)(x+2)=0\). Thus, the critical points are \(x=1\) and \(x=-2\). The intervals \((-rac{ ext{infty}}{\underline{\phantom{xx}}}, -2], [-2, 1], [1, ext{infty})\) need analysis, and the function is increasing on \([2, ext{infty})\). Thus, correctly matched.

Step 4: Analyze Function (C)

The derivative of \(3x^{2}-2x+1\) is \(6x-2\). Setting the derivative to zero gives \(6x-2=0\), solving gives \(x=\frac{1}{3}\). Check intervals: \((-\infty, \frac{1}{3})\) is correct. Therefore, it should be decreasing for \(\left(-\infty, \frac{1}{3}\right]\), therefore is not increasing on this interval.

Step 5: Analyze Function (D)

The derivative of \(x^{3}+6x^{2}+6\) is \(3x^{2}+12x\). Setting the derivative to zero gives \(3x(x+4)=0\), solving gives \(x=0\) or \(x=-4\). Analysis of intervals show the function is increasing on \((-\infty,-4]\), which is correctly matched.

Step 6: Conclusion

Based on the analysis, pair (C) is incorrectly matched as it is stated to be increasing in an interval where it is supposed to be not increasing.

Key Concepts

Increasing Function

An increasing function is a type of function where as you move from left to right along the x-axis, the y-values (or outputs) go up.
In other words, if you choose any two points on the interval and the second point has a greater x-value, its corresponding y-value should also be greater than that of the first point.
This concept is intuitive: just imagine walking uphill as you move forward.

• For a function to be increasing, its derivative needs to be positive over the interval in consideration.
• This means that the slope of the tangent line to the curve of the function is positive, indicating the upward trend.

In the original exercise, to determine whether a function was increasing over a given interval, we needed to carefully analyze its derivative.

Derivative

The derivative of a function is a crucial mathematical tool as it tells us about the function's rate of change at any given point.
Think of it as the function's slope or incline at specific points along its curve.

• Mathematically, the derivative is represented as \(f'(x)\) if \(f(x)\) is the original function.
• A positive derivative indicates the function is increasing, while a negative derivative shows it's decreasing.
• A derivative of zero indicates potential critical points or plateaus where the function might change its behavior from increasing to decreasing or vice versa.

In our original exercise, calculating the derivative was an essential step to finding critical points and understanding how each function behaved over various intervals.

Intervals

Intervals are specific ranges along the x-axis where we analyze the behavior of a function.
For example, intervals are typically expressed in the form \((a, b)\), where \(a\) and \(b\) are the endpoints, and can be open or closed.

• Open intervals \((a, b)\) do not include the endpoints \(a\) and \(b\).
• Closed intervals \([a, b]\) include both endpoints.
• When determining if a function is increasing on an interval, keep in mind that the entire interval's derivative should be positive.

In the exercise, knowing the endpoints and whether they were included helped us confirm if a function was truly increasing over the specified range.

Function Behavior Analysis

Analyzing the behavior of a function is like solving a mystery.
We look for clues along the function's curve, mainly through its derivative, to understand how it changes across intervals.
Critical points, where the derivative equals zero, are valuable pieces of this puzzle.

• These critical points suggest places where the function can change its increasing or decreasing nature.
• The behavior analysis involves checking adjacent intervals and seeing if they follow the expected increasing or decreasing pattern.
• Comparing this behavior with given conditions or constraints in problems helps in identifying mismatches or correctly matched intervals.

In the original task, function behavior analysis allowed us to deduce which intervals were being correctly applied to the function's increasing nature, solving the quiz's central query.