Question

For function \(f(x) = x^{2/3(2x + 5)\), if:}
a) \(x = 0\) is a horizontal tangent
b) The curve has no asymptote
c) The curve has a cusp at (0, 0)
d) The curve has no vertical tangent
Then:

• A. (a), (b) and (c) hold true
• B. Only (a) and (b) hold true
• C. Only (c) is false
• D. (b), (c) and (d) hold true

Hint:

Exam Tip:
For functions with fractional exponents:

• Check the derivative at points where the exponent is negative.
• A cusp occurs when the left and right derivatives have opposite signs.
• A vertical tangent occurs when the derivative tends to the same infinity from both sides.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to analyze the function \(f(x) = x^{2/3}(2x + 5)\) for tangents, asymptotes, cusps, and vertical tangents.

Step 2: Detailed Explanation:
\(f(x) = 2x^{5/3} + 5x^{2/3}\).
• (a) \(x = 0\) is a horizontal tangent:
\(f'(x) = 2 \cdot \frac{5}{3} x^{2/3} + 5 \cdot \frac{2}{3} x^{-1/3} = \frac{10}{3} x^{2/3} + \frac{10}{3} x^{-1/3} = \frac{10}{3} x^{-1/3}(x + 1)\).
As \(x \to 0\), \(f'(x) \to \infty\) (since \(x^{-1/3}\) goes to infinity).
So, the tangent at \(x = 0\) is vertical, not horizontal.
Wait, let's compute \(f'(0)\) more carefully:
\(f'(x) = \frac{10}{3} x^{-1/3}(x + 1)\).
At \(x = 0\), \(x^{-1/3}\) is undefined.
So, the derivative is infinite, indicating a vertical tangent or cusp.
So, (a) is false.
• (b) The curve has no asymptote:
The function is defined for all real \(x\) (since \(x^{2/3}\) is defined for all \(x\)).
As \(x \to \pm \infty\), \(f(x) \to \pm \infty\).
There are no vertical asymptotes and no horizontal asymptotes.
So, (b) is true.
• (c) The curve has a cusp at (0, 0):
At \(x = 0\), \(f(0) = 0\).
The left and right derivatives:
For \(x \to 0^+\), \(f'(x) = \frac{10}{3} x^{-1/3}(x + 1) \to +\infty\).
For \(x \to 0^-\), \(x^{-1/3}\) is negative, so \(f'(x) \to -\infty\).
The left and right derivatives have opposite signs (one \(+\infty\), one \(-\infty\)), which indicates a cusp.
So, (c) is true.
• (d) The curve has no vertical tangent:
A vertical tangent occurs when the derivative tends to the same infinity from both sides.
Here, the derivative tends to \(+\infty\) from the right and \(-\infty\) from the left, which is a cusp, not a vertical tangent.
So, (d) is true (it has no vertical tangent; it has a cusp). So, (a) is false, (b) is true, (c) is true, (d) is true.
Thus, (b), (c), and (d) hold true.
This matches option (D).

Step 4: Final Answer:
Therefore, option (D) is correct.