Question

Use the function `f(x) = sin(x)+cos(x)`, where `0 <= x <= pi`, to answer the following. Help!?

A) Determine all stationary points. Classify the points as relative minima or maxima.
B) Locate any points of inflection.
C) Determine the endpoints of the interval and sketch the graph.

Answer 1

`f(x) = sinx+cosx` for `x in [0,pi]`.

Stationary points are by definitions the points where: `f'(x) = 0`

`f'(x) = cosx-sinx = 0`

`cosx = sinx`

In the interval `[0,pi]` the only value of `x` for which this holds is `x=pi/4`

As:

`f''(x) = -sinx-cosx`

`f''(pi/4) = -sin(pi/4) -cos(pi/4) = -sqrt2 < 0`

the point is a local maximum.

To find points of inflections solve the equation:

`f''(x) = 0`

#-cosx -sinx =0

`sinx = -cosx`

In the interval `[0,pi]` the only value of `x` for which this holds is `x=(3pi)/4`

At the limits of the interval:

`f(0) = 1`

`f(pi) = -1`

To sketch the function however we can note that:

`sinx + cosx = sqrt2(sqrt2/2cosx+sqrt2/2sinx) = sqrt2(cos(pi/4)cosx+sin(pi/4)sinx) = sqrt2 cos(x-pi/4)`

Thus `f(x)` is simply a sinusoid of amplitude `sqrt2` and phase `-pi/4`

graph{sinx+cosx [-0.01, 3.15, -2, 2]}