1. ## Point of Inflexion

Hello!
I have tried to solve this question about the function f(x) = cos x + sin x
But it was a bit difficult for me.
The question is:
Consider the function f(x) = cos x + sin x

a)
(i) Show that f(– p/4) = 0.
(ii) Find in terms of p, the smallest positive value of x which satisfies f(x) = 0.

b) The diagram shows the graph of y = ex (cos x + sin x), – 2 £ x £ 3. The graph has a maximum turning point at C(a, b) and a point of inflexion at D.
(I have attached the diagram)

(i) find dy/dx
(ii) Find the exact value of a and of b.
(iii) Show that at D, y = sqrt2 e^ p/4

Thank you very much for your help!
Best wishes,
Amine

2. a) We may remember that if $\displaystyle \theta$ measures the arc distance from a point on the unit circle counterclockwise from $\displaystyle (x,y)=(1,0)$, then $\displaystyle \cos \theta$ gives the $\displaystyle x$-coordinate of that point and $\displaystyle \sin \theta$ gives the $\displaystyle y$-coordinate.

Therefore, $\displaystyle \cos \theta+\sin \theta = 0$ when $\displaystyle \cos \theta=-\sin \theta$, i.e, when the $\displaystyle y$-coordinate of the point on the circle is the negative of its $\displaystyle x$-coordinate. What fraction of the total arc length, $\displaystyle 2\pi$, do these points give?

b) Points of inflection are found where $\displaystyle y''=0$ and $\displaystyle y'$ differs in sign on both sides next to the point. Extrema are found at critical points: in this case, as $\displaystyle y$ is differentiable and has no boundary points, at points where $\displaystyle y'=0$. To differentiate

$\displaystyle y=e^x(\cos x + \sin x),$

we apply the Product Rule:

$\displaystyle \frac{dy}{dx}=\left(\frac{d}{dx}e^x\right)(\cos x+\sin x)+e^x\left(\frac{d}{dx}(\cos x+\sin x)\right).$

When is $\displaystyle y'=0$? When is $\displaystyle y''=0$? Hint: $\displaystyle e^x$ is always positive.

Hope this helps!