4. (a) Using calculus, find the exact coordinates of the turning points on the curve with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 5 - 2013 - Paper 7

Starting from the original equation: $25x^2 e^x - 16 = 0$

1. Rearranging gives: $25x^2 e^x = 16$
2. Dividing both sides by 25: $x^2 e^x = \frac{16}{25}$
3. Recognizing that this can be rewritten: $x = \frac{4}{5} e^{-x}$

Using the iteration formula:

1. For n=0: $x_1 = \frac{4}{5} e^{-0.5} \approx 0.490$
2. For n=1: $x_2 = \frac{4}{5} e^{-0.490} \approx 0.492$
3. For n=2: $x_3 = \frac{4}{5} e^{-0.492} \approx 0.489$

Thus, rounding each value, we get:

• x_1 = 0.490
• x_2 = 0.492
• x_3 = 0.489

From our previous iterations, we see that the values are converging. We have:

• x_2 = 0.492
• x_3 = 0.489

Analyzing these results indicates that \alpha can be estimated to be approximately 0.49 when rounded to two decimal places.

We can justify this by observing that the values are approaching 0.49, which suggests it is a good approximation of the root.