Problem 10 - Calculus for JEE ADVANCED

Question

$$ \begin{aligned} &f(x)=e^{x^{3}-3 x+2} \\ &\text { Let } g(x)=x^{3}-3 x+2 \\ &\quad g^{\prime}(x)=3 x^{2}-3=3\left(x^{2}-1\right) \\ &\geq 0 \text { for } x \in(-\infty,-1] \end{aligned} $$ Now, the range of \(f(x)\) is \(\left(0, e^{4}\right]\) But co-domain is \(\left(0, e^{5}\right]\) Hence, \(f(x)\) is an into function.

Step-by-Step Solution

Answer

The range of \( f(x) \) is \((0, e^4]\) and it is an into function.

Step 1: Understanding the Functions

The given function is \( f(x) = e^{x^3 - 3x + 2} \). It is defined in terms of another function, \( g(x) = x^3 - 3x + 2 \). We need to examine \( g(x) \) because it affects the behavior and range of \( f(x) \).

Step 2: Finding the Nature of g(x)

The derivative of \( g(x) \) is given as \( g'(x) = 3(x^2 - 1) \). This can be factored and analyzed as \( 3(x+1)(x-1) \). This indicates that \( g(x) \) changes its behavior at \( x = 1 \) and \( x = -1 \).

Step 3: Analyzing g(x) for Monotonicity

From \( g'(x) \), we know \( g(x) \) is increasing in intervals \((-∞,-1] \) and \([1, ∞) \) and decreasing in \( (-1, 1) \). For \( f(x) = e^{g(x)} \), note that \( f(x) \) will take its minimum and maximum values in these intervals where \( g(x) \) has its minimum and maximum.

Step 4: Determine Range Values of g(x) and f(x)

When \( x = -1 \), \( g(x) = (-1)^3 - 3(-1) + 2 = 4 \), the maximum value when \( x \leq -1 \). At \( x = 1 \), \( g(x) = (1)^3 - 3(1) + 2 = 0 \), a minimum point. For \( f(x) = e^{g(x)} \), the range is \((0, e^4]\).

Step 5: Understanding Co-domain

The co-domain is given as \((0, e^5]\). Compare this with the range of \( f(x) \) which is \((0, e^4]\). Since the range is a subset of the co-domain and doesn't cover it completely, \( f(x) \) is indeed an "into" function.

Key Concepts

Monotonicity

Monotonicity describes how a function behaves as it moves through its input values. Specifically, whether it consistently increases or decreases. It's important because it helps us understand how functions change and find their extrema. In this exercise, we determine the monotonicity of \(g(x) = x^3 - 3x + 2\) by examining its derivative. To understand monotonicity:

• A function is increasing where its first derivative is positive.
• A function is decreasing where its first derivative is negative.

For \(g(x)\), the derivative \(g'(x) = 3(x^2 - 1)\) is analyzed:

• It is positive when \(x < -1\) and \(x > 1\), meaning \(g(x)\) is increasing.
• It is negative between \(-1 < x < 1\), meaning \(g(x)\) is decreasing.

This helps us determine where the maximum and minimum values of \(g(x)\) occur, thereby impacting the range of our exponential function \(f(x) = e^{g(x)}\).

Derivative Analysis

Derivative analysis involves computing the first derivative of a function to gain insights about its behavior, such as monotonicity and identifying critical points. In this case, we analyze \(g(x) = x^3 - 3x + 2\). The derivative \(g'(x) = 3(x^2 - 1) = 3(x+1)(x-1)\) tells us where \(g(x)\) changes its monotonicity. Points where the derivative equals zero, \(x = -1\) and \(x = 1\), are called critical points. Using this information:

• Increasing intervals: We determine that \(g(x)\) is increasing on \((-\infty, -1]\) and \([1, \infty)\).
• Decreasing interval: It decreases on \((-1, 1)\).

Understanding these changes provides insight into where the function reaches minimum and maximum values, crucial for mapping the range of the main function \(f(x) = e^{g(x)}\).

Exponential Function

An exponential function is one where the variable \((x)\) appears in the exponent. These functions are known for their rapid growth or decay, depending on their base. Our function of interest is \(f(x) = e^{g(x)}\), where \(e\) is Euler's number, approximately equal to \(2.718\).What makes exponential functions unique is their properties:

• They are always increasing if the base is greater than one, like in this case.
• They exhibit swift changes in value, reflective of changes in their exponents.

For \(f(x) = e^{g(x)}\), we use the behavior of \(g(x)\) explored through monotonicity and derivative analysis:

• If \(g(x)\) increases or reaches a peak, \(f(x)\) sharply increases accordingly.
• If \(g(x)\) decreases, \(f(x)\) also decreases but remains above zero.

The range \((0, e^4]\) for \(f(x)\) is derived from the maximum and minimum values within the intervals determined by the monotonicity of \(g(x)\). This systematic approach ensures we accurately capture how \(g(x)\) affects \(f(x)\).