The function, f(x) = (3x - 7)x^(2/3), x ∈ R, is increasing for all x lying in : (A) (-∞,0) ∪ (14/15,

Q: Wait, why is $x=0$ a critical point? The function is defined there!

That is a fantastic observation! A critical point isn't just where the derivative is zero; it is also where the derivative fails to exist. At $x=0$, the derivative $f'(x)$ involves $x^{1/3}$ in the denominator, which causes the expression to be undefined. Geometrically, this means the tangent line becomes vertical at the origin. Since the slope changes behavior there, we absolutely must include it as a critical point.

Q: Why do we use the union symbol $\cup$ in the answer?

The union symbol is our mathematical way of saying 'both these regions are valid.' We found two separate intervals on the number line where the function is climbing uphill. Since both regions satisfy the condition $f'(x) > 0$, we combine them into one set. It is like saying 'I am happy in this room OR in that room'—the union connects them into one complete solution.

Q: Can I just ignore the denominator when checking signs?

Absolutely not! The denominator is just as important as the numerator. If you ignore it, you will miss the sign flip at $x=0$. Always treat the derivative as a single rational expression before testing intervals. The sign of the entire fraction depends on both the numerator AND the denominator.

Question

The function, $f(x) = (3x - 7)x^{2/3}$, $x \in \mathbb{R}$, is increasing for all $x$ lying in :

Options:

(A) $(-\infty,0) \cup \left(\frac{14}{15},\infty\right)$

(B) $(-\infty, 0) \cup \left(\frac{7}{8},\infty\right)$

(C) $(-\infty,-\frac{14}{15}) \cup (0,\infty)$

(D) $(-\infty, \frac{14}{15})$

Sigma Percentile · Accepted Answer

### The Mountaineer’s Guide to Calculus: Understanding the Slope Imagine you are standing at the base of a mountain range. You are looking at the path of a function, $f(x) = (3x - 7)x^{2/3}$. Your goal is to determine exactly where you are walking uphill. In the language of calculus, we are looking for the intervals where the function is strictly increasing, which occurs where the slope of the tangent line is positive, or mathematically, where $f'(x) > 0$. ### Phase 1: The Product Rule Dance To find the slope, we need the derivative. Since our function is a product of two entities, $(3x - 7)$ and $x^{2/3}$, we apply the Product Rule: $\frac{d}{dx}[u \cdot v] = u'v + uv'$. Let $u = (3x - 7)$ and $v = x^{2/3}$. The derivative of $u$ is $\frac{d}{dx}(3x - 7) = 3$. The derivative of $v$ follows the power rule: $\frac{d}{dx}(x^{2/3}) = \frac{2}{3}x^{-1/3}$. Assembling these pieces, we get: $$f'(x) = 3 \cdot x^{2/3} + (3x - 7) \cdot \frac{2}{3}x^{-1/3}$$ This expression can be rewritten to clarify its structure: $$f'(x) = 3x^{2/3} + \frac{2(3x - 7)}{3x^{1/3}}$$ ### Phase 2: The Art of Simplification To analyze the sign of the derivative, we must combine these terms into a single fraction using the common denominator $3x^{1/3}$. Multiplying the first term by $\frac{3x^{1/3}}{3x^{1/3}}$, we obtain: $$f'(x) = \frac{3x^{2/3} \cdot 3x^{1/3} + 2(3x - 7)}{3x^{1/3}}$$ Since $x^{2/3} \cdot x^{1/3} = x^1$, the expression simplifies significantly: $$f'(x) = \frac{9x + 6x - 14}{3x^{1/3}} = \frac{15x - 14}{3x^{1/3}}$$ ### Phase 3: The Gatekeepers (Critical Points) Critical points occur where the derivative is zero or undefined. These points define the boundaries where the function changes behavior. 1. Setting the numerator to zero: $15x - 14 = 0 \Rightarrow x = \frac{14}{15}$. 2. Setting the denominator to zero: $3x^{1/3} = 0 \Rightarrow x = 0$. These points, $0$ and $\frac{14}{15}$, divide the real number line into three distinct territories. We must test each to determine if the slope is positive or negative. ### Phase 4: The Number Line Dance **Region 1: $x < 0$** Pick $x = -1$. The numerator $15(-1) - 14 = -29$ is negative, and the denominator $3(-1)^{1/3} = -3$ is negative. Since a negative divided by a negative is positive, the function is **increasing**. **Region 2: $0 < x < \frac{14}{15}$** Pick $x = 0.5$. The numerator $15(0.5) - 14 = -6.5$ is negative, while the denominator $3(0.5)^{1/3}$ is positive. A negative divided by a positive is negative, so the function is **decreasing**. **Region 3: $x > \frac{14}{15}$** Pick $x = 1$. The numerator $15(1) - 14 = 1$ is positive, and the denominator $3(1)^{1/3} = 3$ is positive. A positive divided by a positive is positive, so the function is **increasing**. ### The Conclusion We have successfully mapped the terrain. The function is strictly increasing in the intervals $(-\infty, 0)$ and $(\frac{14}{15}, \infty)$. Combining these, the final answer is: $$(-\infty, 0) \cup \left(\frac{14}{15}, \infty\right)$$

Animated Solution for Mathematics - Differentiation: The function, $f (x) = (3 x - 7) x^{2/3}$ , $x \in R$ , is increasing for all $x$ lying in :

Select Answer:

Visualized Solution

Analyze the Function $f (x)$

Condition for Increasing Function

Applying the Product Rule

Differentiating the Terms

Simplifying the Derivative

Final Form of $f^{'} (x)$

Finding Critical Points

Sign Analysis: $x < 0$

Sign Analysis: $0 < x < \frac{14}{15}$

Sign Analysis: $x > \frac{14}{15}$

Conclusion & Takeaway

The Sigma Insight: Monotonicity

The Mountaineer’s Guide to Calculus

Phase 1

Phase 2

Phase 3

Phase 4

The Conclusion

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Animated Solution for Mathematics - Differentiation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function, f(x)=(3x−7)x2/3, x∈R, is increasing for all x lying in :