Problem 130 - 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

(A) \(h\) is increasing whenever \(f\) is increasing.

Step 1: Define the derivative of the given function

The function is given as \( h(x) = f(x) - [f(x)]^2 + [f(x)]^3 \). To find out when \( h(x) \) is increasing or decreasing, we need to determine its derivative, \( h'(x) \). Use the chain rule and the power rule for differentiation. We have: \[ h'(x) = \frac{d}{dx}(f(x)) - \frac{d}{dx}([f(x)]^2) + \frac{d}{dx}([f(x)]^3) \]Each of these derivatives can be calculated as follows.

Step 2: Differentiate each term

Differentiate each term separately for \(h'(x)\): 1. \(\frac{d}{dx} f(x) = f'(x)\)2. \(\frac{d}{dx} [f(x)]^2 = 2f(x)f'(x)\)3. \(\frac{d}{dx} [f(x)]^3 = 3[f(x)]^2f'(x)\)Substitute these into the derivative of \(h(x)\):\[ h'(x) = f'(x) - 2f(x)f'(x) + 3[f(x)]^2f'(x) \].

Step 3: Factor the derivative

Factor \(h'(x)\) to determine the effect of \(f'(x)\) on \(h(x)\):\[ h'(x) = f'(x)(1 - 2f(x) + 3[f(x)]^2) \].The term \(1 - 2f(x) + 3[f(x)]^2 \) determines the sign of \(h'(x)\). Now, simplify \(1 - 2f(x) + 3[f(x)]^2\): Let it be \(g(f(x)) = 3[f(x)]^2 - 2f(x) + 1\).

Step 4: Analyze the sign of the factored expression

The quadratic expression \(g(t) = 3t^2 - 2t + 1\) is always positive, since it is a parabola opening upwards (leading coefficient is 3). The discriminant of \(3t^2 - 2t + 1\) is negative (\(b^2 - 4ac = (-2)^2 - 4 \times 3 \times 1 = -8\)), confirming that it has no real roots and is always positive. Therefore, \(h'(x) = f'(x)\cdot g(t)\), and \(f'(x)\) essentially drives the behavior of \(h(x)\).

Step 5: Conclude the behavior of h depending on f

Since \(g(t)\) is always positive, \(h'(x) = f'(x)\cdot g(f(x))\) simplifies to the sign of \(f'(x)\). Therefore, if \(f(x)\) is increasing (\(f'(x) > 0\)), then \(h(x)\) is also increasing. If \(f(x)\) is decreasing (\(f'(x) < 0\)), then \(h(x)\) is decreasing.

Key Concepts

Derivatives

In calculus, the derivative of a function helps us understand how the function's outputs change in response to changes in inputs. It represents the rate of change or slope of the function at any given point. For a function like \( h(x) \), which is dependent on another function \( f(x) \), the derivative is crucial to determining its increasing or decreasing behavior.

To find the derivative \( h'(x) \) of our given function \( h(x) = f(x) - [f(x)]^2 + [f(x)]^3 \), we use the sum of derivatives for each part of the function. This involves calculating the derivative of each term separately:

• For \( f(x) \), the derivative is simply \( f'(x) \).
• The derivative of \( [f(x)]^2 \) is \( 2f(x)f'(x) \), using the chain rule.
• The derivative of \( [f(x)]^3 \) is \( 3[f(x)]^2f'(x) \), again applying the chain rule.

When these are combined, we obtain \( h'(x) = f'(x) - 2f(x)f'(x) + 3[f(x)]^2f'(x) \). Understanding this result, \( h'(x) \), allows us to explore when the function is increasing or decreasing.

Increasing and Decreasing Functions

Increasing and decreasing functions are determined by the sign of their derivatives.

• If the derivative \( h'(x) \) is positive, the function is increasing. This means as \( x \) increases, \( h(x) \) increases.
• If \( h'(x) \) is negative, the function is decreasing, which implies \( h(x) \) decreases as \( x \) goes up.
• If \( h'(x) = 0 \), the function is constant at that interval.

In our function \( h(x) \), we've discovered that \( h'(x) = f'(x)(1 - 2f(x) + 3[f(x)]^2) \). By analyzing \( 1 - 2f(x) + 3[f(x)]^2 \), we find it is always positive. This means the overall sign of \( h'(x) \) relies entirely on \( f'(x) \). Therefore, if \( f(x) \) is increasing (when \( f'(x) > 0 \)), \( h(x) \) will also be increasing. Conversely, if \( f(x) \) is decreasing (\( f'(x) < 0 \)), \( h(x) \) will decrease.

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions.

When you have a function that is composed of other functions, like \( h(x) = f(x) - [f(x)]^2 + [f(x)]^3 \), the chain rule helps differentiate each term. It states that if you have a composite function \( y = g(u) \), where \( u = f(x) \), the derivative \( \frac{dy}{dx} \) is \( \frac{dg}{du} \times \frac{du}{dx} \).

In our exercise, we applied the chain rule as follows:

• For \([f(x)]^2\), we let \( u = f(x) \), so its derivative becomes \( 2u \times u' = 2f(x)f'(x) \).
• For \([f(x)]^3\), similarly, the chain rule gives us \( 3[u]^2 \times u' = 3[f(x)]^2f'(x) \).

These applications allow us to systematically calculate the derivatives of composite functions, giving us the necessary tools to understand their behavior in terms of increasing and decreasing intervals. The chain rule simplifies the process of finding derivatives for complex functions, making it an essential part of calculus.