Problem 121 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(h(x)=f(x)-[f(x)]^{2}+[f(x)]^{3}\) for every real number \(x\). Then (A) \(h\) is increasing whenever \(f\) is increasing (B) \(h\) is increasing whenever \(f\) is decreasing (C) \(h\) is decreasing whenever \(f\) is decreasing (D) nothing can be said in general

Step-by-Step Solution

Answer

(D) Nothing can be said in general.

Step 1: Express the Derivative of h(x)

The function is given as \( h(x) = f(x) - [f(x)]^{2} + [f(x)]^{3} \). To analyze whether \( h \) is increasing or decreasing, we need to find its derivative, \( h'(x) \). It can be expressed as: \[ h(x) = f(x)(1 - f(x) + [f(x)]^{2}) \]. Therefore, we use the product and chain rule to find the derivative: \[ h'(x) = f'(x) \cdot (1 - 2f(x) + 3[f(x)]^2) \].

Step 2: Analyze the Sign of the Derivative

Since \( h'(x) = f'(x) \cdot (1 - 2f(x) + 3[f(x)]^2) \), the sign of \( h'(x) \) depends on both \( f'(x) \) and \( 1 - 2f(x) + 3[f(x)]^2 \). We need to check for values of \( f(x) \) where this expression might change sign.

Step 3: Consider Different Cases for f(x)

Evaluate \( 1 - 2f(x) + 3[f(x)]^2 \) to determine if it changes sign.- For small values, assume \( f(x) = 0 \), \(1 \ast 0 + 3[0]^2 = 1\), which is positive.- As \( f(x)\) moves toward 1, calculate at \( f(x) = 1 \), \(1 \ast 1+3[1]^2-2 \ast 1 = 2 \), still positive.

Step 4: Conclusion for h(x)

Since \( h'(x) = f'(x) \cdot (1 - 2f(x) + 3[f(x)]^2) \) the sign at most \( f(x) \), it always remains positive, leading to inconsistency when \(f'(x) \leq 0 \). Similarly, if \( f(x) \) is decreasing \( f'(x) < 0 \). There is no general behavior between \(h(x)\) as whether \( f(x) \) increases or decreases.

Key Concepts

Derivative

In calculus, derivatives play a crucial role in understanding how a function behaves. For a function like \( h(x) = f(x) - [f(x)]^{2} + [f(x)]^{3} \), the derivative, written as \( h'(x) \), gives us the rate at which \( h(x) \) changes as \( x \) changes.
To find the derivative of a function that is a combination of terms like this, we must use rules such as the product and chain rules. The derivative is calculated as:

• \( h'(x) = f'(x) \cdot (1 - 2f(x) + 3[f(x)]^2) \),

This tells us how \( h(x) \) responds to small changes in \( x \). Understanding the derivative helps in deciding whether \( h(x) \) is increasing or decreasing.
By analyzing \( h'(x) \), we gain insight into the behavior of the original function \( h(x) \), especially its trends or shifts.

Function Analysis

Function analysis involves dissecting the behavior and characteristics of a given function. For \( h(x) \), we analyze \( h'(x) = f'(x) \cdot (1 - 2f(x) + 3[f(x)]^2) \). Here, the expression \( 1 - 2f(x) + 3[f(x)]^2 \) is especially critical in determining the overall trend of the function.
To understand this fully, we examine:

• The sign of \( f'(x) \), which tells us if \( f(x) \) is increasing (when \( f'(x) > 0 \)) or decreasing (when \( f'(x) < 0 \)).
• The expression \( 1 - 2f(x) + 3[f(x)]^2 \) remains positive in specific intervals. Exploring this can help predict if \( h(x) \) increases or decreases depending on changes in \( f(x) \).

By performing this type of analysis, we can make more informed conclusions about how the function behaves over its domain.

Behavior of Functions

The behavior of a function, like \( h(x) \), is crucial in determining how it changes over different ranges of \( x \). To explore this for \( h(x) \), we look at the sign of its derivative \( h'(x) \). Understanding whether this derivative is positive, negative, or zero can tell us:

• When \( h(x) \) is increasing or decreasing. If \( f(x) \) is increasing and \( 1 - 2f(x) + 3[f(x)]^2 \) maintains a positive sign, \( h(x) \) will also increase if \( f'(x) > 0 \).
• Unique behaviors or exceptions. There is an inconsistency in the changes in \( h(x) \) when \( f(x) \) decreases if \( f'(x) < 0 \). Such exceptions demonstrate that sometimes no specific general behavior can be ascertained without more data.

This method showcases how critical it is to examine derivatives closely since they provide valuable insights into the character and tendencies of a function across various intervals of \( x \).