The curve C has equation y = 3x^3 - 8x^2 - 3 (a) Find (i) \( \frac{dy}{dx} \) (ii) \( \frac{d^2y}{dx^2} \) (b) Verify that C has a stationary point when \( x = 2 \) (c) Determine the nature of this stationary point, giving a reason for your answer. - Edexcel - A-Level Maths Pure - Question 3 - 2017 - Paper 1

Now, we differentiate ( \frac{dy}{dx} ) to find ( \frac{d^2y}{dx^2} ):

[ \frac{dy}{dx} = 9x^2 - 16x ]

Differentiating again:

[ \frac{d^2y}{dx^2} = 18x - 16 ]

To verify if there is a stationary point at ( x = 2 ), substitute into ( \frac{dy}{dx} ):

[ \frac{dy}{dx} = 9(2)^2 - 16(2) = 36 - 32 = 4 ]

Since ( \frac{dy}{dx} = 0 ) is not satisfied (it's 4), this indicates that there is no stationary point at ( x = 2 ). Hence, more assessments at nearby values may be required.

As concluded in part (b), since the first derivative ( \frac{dy}{dx} ) at ( x = 2 ) does not equal zero, hence, there is no stationary point at that value. Therefore, there can be no nature to determine since the derivative does not yield a stationary value.