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education/math/MATH1210 (calc 1)/Max and Min.md
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@ -34,4 +34,11 @@ If $f$ is a continuous function in a closed interval $[a, b]$, then $f$ achieves
> Find the absolute maximum and absolute minimum of the function $h(x) = x + 2cos(x)$ on the closed interval $[0, \pi]$. > Find the absolute maximum and absolute minimum of the function $h(x) = x + 2cos(x)$ on the closed interval $[0, \pi]$.
1. $x = 0$ and $x = \pi$ are both critical numbers because they are endpoints. Endpoints are critical because $\dfrac{d}{dx}$ is undefined. 1. $x = 0$ and $x = \pi$ are both critical numbers because they are endpoints. Endpoints are critical because $\dfrac{d}{dx}$ is undefined.
2. 2. $\dfrac{d}{dx} x + 2\cos(x) = 1 - 2sin(x)$
3. Setting that to zero, we get $0 = 1 - 2\sin(x)$
4. $\sin(x) = \dfrac{1}{2}$
5. In the interval $[0, \pi]$, $\sin(x)$ has a value of $\dfrac{1}{2}$ in two places: $x = \dfrac{\pi}{6}$ and $x = \dfrac{5\pi}{6}$. These are both critical numbers because they are points where $\dfrac{d}{dx}$ is zero.
6. Now we plug these values into the original function:
7. $h(0) = 0 + 2\cos 0 = 2$
8. $h(\pi) = \pi + 2\cos(\pi) = \pi - 2$
9. $h(\dfrac{pi}{6}) = \dfrac{\pi}{6} + 2\cos(\dfrac{pi}$