Key Concepts and Formulas

• Monotonicity using the First Derivative: A function $f(x)$ is strictly increasing on an interval if $f'(x) > 0$ for all $x$ in the interval, and strictly decreasing if $f'(x) < 0$ for all $x$ in the interval.
• Power Rule for Differentiation: $\frac{d}{dx}(x^n) = nx^{n-1}$
• Discriminant of a Quadratic: For a quadratic equation $ax^2 + bx + c = 0$, the discriminant is given by $\Delta = b^2 - 4ac$. If $\Delta < 0$, the quadratic has no real roots and its sign is the same as the sign of $a$.

Step-by-Step Solution

1. Understand the Given Function and the Goal The given function is $f(x) = x^3 - 3x^2 + 5x + 7$. Our goal is to determine the intervals where this function is increasing or decreasing.

2. Step 1: Calculate the First Derivative of the Function To analyze the monotonicity of $f(x)$, we need to find its first derivative, $f'(x)$. Using the power rule and sum/difference rule for differentiation: $f'(x) = \frac{d}{dx}(x^3 - 3x^2 + 5x + 7)$ $f'(x) = \frac{d}{dx}(x^3) - 3\frac{d}{dx}(x^2) + 5\frac{d}{dx}(x) + \frac{d}{dx}(7)$ $f'(x) = 3x^2 - 3(2x) + 5(1) + 0$ $f'(x) = 3x^2 - 6x + 5$ So, the first derivative is $f'(x) = 3x^2 - 6x + 5$.

3. Step 2: Analyze the Sign of the First Derivative We need to determine the sign of the quadratic expression $f'(x) = 3x^2 - 6x + 5$. The leading coefficient is $A = 3$, which is positive, meaning the parabola opens upwards. The discriminant is $\Delta = B^2 - 4AC = (-6)^2 - 4(3)(5) = 36 - 60 = -24$. Since the discriminant is negative ($\Delta = -24 < 0$) and the leading coefficient is positive ($A = 3 > 0$), the quadratic expression $f'(x) = 3x^2 - 6x + 5$ is always positive for all real values of $x$. This is because the parabola never intersects the x-axis and opens upwards, so it is always above the x-axis. $f'(x) > 0 \quad \text{for all } x \in \mathbb{R}$

4. Step 3: Conclude the Monotonicity of the Function Since $f'(x) > 0$ for all $x \in \mathbb{R}$, the function $f(x)$ is strictly increasing on $\mathbb{R}$.

5. Compare with Options

• (A) increasing in R . - This matches our conclusion.
• (B) decreasing in R . - Incorrect, as $f'(x) > 0$.
• (C) decreasing in (0, $\infty$) and increasing in ($-$\infty$$, 0) - Incorrect, as $f'(x)$ is always positive.
• (D) increasing in (0, $\infty$) and decreasing in ($-$\infty$$, 0) - Incorrect, as $f'(x)$ is always positive.

Thus, the correct option is (A).

Common Mistakes & Tips

• Careless Differentiation: Double-check your derivative calculation. Mistakes in applying the power rule or constant multiple rule can lead to incorrect results.
• Incorrectly Analyzing the Quadratic: Remember to consider both the leading coefficient and the discriminant when determining the sign of a quadratic expression. A negative discriminant implies no real roots, and the sign of the quadratic is determined by the leading coefficient.
• Confusing Increasing/Decreasing: Make sure you understand the relationship between the sign of the first derivative and the monotonicity of the function. $f'(x) > 0$ implies increasing, and $f'(x) < 0$ implies decreasing.

Summary

To determine the monotonicity of the given function $f(x) = x^3 - 3x^2 + 5x + 7$, we calculated its first derivative $f'(x) = 3x^2 - 6x + 5$. By analyzing the discriminant of the quadratic expression for $f'(x)$, we found that $f'(x) > 0$ for all real values of $x$. Therefore, the function $f(x)$ is strictly increasing on $\mathbb{R}$, which corresponds to option (A).

Final Answer The final answer is \boxed{increasing in R}, which corresponds to option (A).