A curve C has equation y = e^{x} + x^{3} + 8x + 5 (a) Show that the x coordinate of any turning point of C satisfies the equation x^{2} = 2 - e^{x} (b) On the axes given on page 5, sketch, on a single diagram, the curves with equations (i) y = x^{3} (ii) y = 2 - e^{x} On your diagram give the coordinates of the points where each curve crosses the y-axis and state the equation of any asymptotes - Edexcel - A-Level Maths Pure - Question 4 - 2014 - Paper 6

i) For the curve y = x^{3}, it must pass through the origin and has a characteristic S-shape, starting in the third quadrant, rising through the origin to the first quadrant.

ii) The curve y = 2 - e^{x} needs careful consideration. This curve intersects the y-axis at (0, 2) since: $y = 2 - e^{0} = 1$. The horizontal asymptote occurs when x approaches infinity, where the curve approaches the line y = 2.

In the same diagram, plot both curves with clear labeling, showing their intersections and crossing points.

From the plotted curves, it is evident that the curve y = x^{2} intersects the curve y = 2 - e^{x} at only one point. This illustrates that the equation x^{2} = 2 - e^{x} has a unique solution since the two curves touch at a single location, which indicates that there is exactly one crossing point.

Using the iteration formula

$x_{n+1} = (-2 - e^{-x_{n}})^{\frac{1}{3}}$

Starting with $x_{0} = -1$, we compute:

1. For $n=0$: $x_{1} = (-2 - e^{-(-1)})^{\frac{1}{3}}$ This gives approximately $x_{1} = -1.26376.$

2. For $n=1$: $x_{2} = (-2 - e^{-(-1.26376)})^{\frac{1}{3}}$ This results in approximately $x_{2} = -1.26126.$

Both values rounded to 5 decimal places are:

• $x_{1} = -1.26376$
• $x_{2} = -1.26126$.

From our calculations, we find the approximate root of the x-coordinates at:

$x \approx -1.26.$

Next, substituting this x-value back into the original function to find the corresponding y-coordinate:

$y = e^{-1.26} + (-1.26)^{3} + 8(-1.26) + 5.$

After computing, this gives approximately: $y = 0.24.$

Thus, the coordinates of the turning point of the curve C are (approximately) $(-1.26, 0.24)$, which can subsequently be rounded to $(-1.26, 0.24)$ when required to two decimal places.