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: Compute the second derivative of \(y = (x - a)^n\).

\[\frac{d^2y}{dx^2} = n(n - 1)(x - a)^{n - 2}\]

Step 2: To locate inflection points, equate the second derivative to zero:

\[n(n - 1)(x - a)^{n - 2} = 0\]

This yields \(x = a\).

Step 3: Determine the \(n\)-th derivative of \(y = (x - a)^n\):

\[\frac{d^n y}{dx^n} = n!\]

Given that \(n\) is odd, it follows that \( \frac{d^n y}{dx^n} eq 0\) and \( \frac{d^{n-1}y}{dx^{n-1}} = 0\). Consequently, the inflection point is located at \((a, 0)\).