Find the local maxima and local minima of the function f(x) = sin(x) + cos(x) on the interval [0, 2*pi].

Step 1: Find the first derivative

To find local maxima and minima, we first need to find the critical points of the function. Critical points occur where the first derivative is zero or undefined. We start by calculating the first derivative of the given function $f(x) = \sin(x) + \cos(x)$.

Step 2: Find critical points

Next, we set the first derivative equal to zero to find the critical points. This means we are looking for values of $x$ where the tangent function is equal to 1. We solve this equation for $x$ in the given interval $[0, 2\pi]$.

Step 3: Solve for x

In the interval $[0, 2\pi]$, the values of $x$ for which $\tan(x) = 1$ are $x = \frac{\pi}{4}$ and $x = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$. These are our critical points.

Step 4: Find the second derivative

To determine whether these critical points correspond to a local maximum or minimum, we use the second derivative test. We calculate the second derivative of the function.

Step 5: Evaluate second derivative at critical points

We evaluate the second derivative at each critical point. If $f''(x) < 0$, it's a local maximum. If $f''(x) > 0$, it's a local minimum. If $f''(x) = 0$, the test is inconclusive.

Step 6: Calculate function values at critical points

Based on the second derivative test, $x = \frac{\pi}{4}$ corresponds to a local maximum, and $x = \frac{5\pi}{4}$ corresponds to a local minimum. We calculate the function values at these points to find the actual maximum and minimum values.

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