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

To show this, we can start from the equation:

$25x^2 e^x - 16 = 0$

Rearranging this gives:

$25x^2 e^x = 16$

Dividing both sides by 25 gives:

$x^2 e^x = \frac{16}{25}$

Taking the square root on both sides:

$xe^x = \frac{16}{25}$

This can be rewritten as:

$x = \frac{4}{5} e^{x}$

Using the iteration formula:

1. For x_0 = 0.5: $x_1 = \frac{4}{5} e^{0.5} \approx 0.49$

2. For x_1: $x_2 = \frac{4}{5} e^{0.49} \approx 0.492$

3. For x_2: $x_3 = \frac{4}{5} e^{0.492} \approx 0.489$

Thus, the values are: x_1 \approx 0.490, x_2 \approx 0.492, x_3 \approx 0.489.

The estimates from the previous calculation converge around α = 0.49. To justify this accuracy, we can validate the values using the function:

1. Calculate f(0.485), f(0.492), and f(0.495).
2. We find that f(0.49) is approximately 0. Thus, the accurate estimate for α is 0.49.