Question

The point of inflexion for the curve \(y = (x - a)^n\), where \(n\) is odd integer and \(n \ge 3\), is:

• A. \((a, 0)\)
• B. \((0, a)\)
• C. \((0, 0)\)
• D. None of these

Hint:

For curves involving powers, differentiate multiple times and check for points where the second derivative changes sign to find inflection points.

Answer 1

The Correct Option is A

Step 1: Differentiate \(y = (x - a)^n\) to find the second derivative. \[ \frac{d^2y}{dx^2} = n(n - 1)(x - a)^{n - 2} \] Step 2: For the point of inflexion, set \(\frac{d^2y}{dx^2} = 0\): \[ n(n - 1)(x - a)^{n - 2} = 0 \] This gives \(x = a\). Step 3: Now, differentiate \(y = (x - a)^n\) \(n\) times: \[ \frac{d^n y}{dx^n} = n! \] Since \(n\) is odd, we have \( \frac{d^n y}{dx^n} \neq 0\) and \( \frac{d^{n-1}y}{dx^{n-1}} = 0\). Therefore, the point of inflexion is \((a, 0)\).

Answer 2

Step 1: Write the given function
The given curve is \( y = (x - a)^n \), where \( n \) is an odd integer and \( n \ge 3 \).
Step 2: Find the first and second derivatives of \( y \)
The first derivative of \( y \) with respect to \( x \) is: \[ \frac{dy}{dx} = n(x - a)^{n-1}. \] Now, take the second derivative to find the inflection point: \[ \frac{d^2y}{dx^2} = n(n-1)(x - a)^{n-2}. \]
Step 3: Find the condition for an inflection point
An inflection point occurs where the second derivative changes sign, which means \( \frac{d^2y}{dx^2} = 0 \). Set the second derivative equal to zero: \[ n(n-1)(x - a)^{n-2} = 0. \] Since \( n \ge 3 \), \( n(n-1) \neq 0 \), so the equation simplifies to: \[ (x - a)^{n-2} = 0. \] This implies: \[ x - a = 0 \quad \Rightarrow \quad x = a. \]
Step 4: Find the corresponding value of \( y \)
To find the corresponding value of \( y \) at \( x = a \), substitute \( x = a \) into the original equation \( y = (x - a)^n \): \[ y = (a - a)^n = 0. \]
Step 5: Conclusion
Thus, the point of inflection for the curve is \( (a, 0) \). \[ \boxed{(a, 0)}. \]