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 \((a, 0)\).