# How do you find absolute extrema of the function g(x) = 2x + 5cosx on the interval [0,2pi]?

Apr 13, 2015

The minimum is $g \left(\pi - {\sin}^{- 1} \left(\frac{2}{5}\right)\right) \approx 0.8776$
The maximum is $g \left(2 \pi\right) \approx 17.5663$

To find absolute extrema of a function that is continuous on a closed interval: find critical numbers that are in the interval, evaluate the function at the endpoints and at the critical numbers

You'll want some technology to finish the arithmetic.
$g \left(x\right) = 2 x + 5 \cos x$ on the interval $\left[0 , 2 \pi\right]$

$g ' \left(x\right) = 2 - 5 \sin x = 0$ where $\sin x = \frac{2}{5}$

I do not recognize a value of $x$ that will work, but I do know that there are 2 such value in the interval $\left[0 , 2 \pi\right]$. One is in the first quadrant (call it ${x}_{1}$) and one is in the second (${x}_{2}$).

Using ${\sin}^{2} x + {\cos}^{2} x = 1$ we can work out that at ${x}_{1}$ we also have
$\cos {x}_{1} = \sqrt{1 - {\sin}^{2} {x}_{1}} = \sqrt{1 - {\left(\frac{2}{5}\right)}^{2}} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}$.

At the other, $\cos {x}_{2} = - \frac{\sqrt{21}}{5}$.

Now, we need to find $g \left(x\right)$ for four numbers:

$g \left(0\right) = 2 \left(0\right) + 5 \cos \left(0\right) = 5$

$g \left({x}_{1}\right) = 2 \left({x}_{1}\right) + 5 \cos \left({x}_{1}\right) = 2 \left({x}_{1}\right) + 5 \left(\frac{\sqrt{21}}{5}\right) = 2 \left({x}_{1}\right) + \sqrt{21}$

$g \left({x}_{2}\right) = 2 \left({x}_{2}\right) + 5 \cos \left({x}_{2}\right) = 2 \left({x}_{2}\right) + 5 \left(- \frac{\sqrt{21}}{5}\right) = 2 \left({x}_{2}\right) - \sqrt{21}$

$g \left(2 \pi\right) = 2 \left(2 \pi\right) + 5 \cos \left(2 \pi\right) = 4 \pi + 5$

Using technology to find the values of ${x}_{1}$ and ${x}_{2}$

With ${x}_{1} \approx 0.4115$ (so ${x}_{2} = \pi - {x}_{1} \approx 2.7301$)
and $\sqrt{21} \approx 4.5826$,

We get
$g \left(0\right) = = 5$
$g \left({x}_{1}\right) \approx 5.4056$
$g \left({x}_{2}\right) \approx 0.8776$
$g \left(2 \pi\right) \approx 17.5663$

The minimum is $g \left({x}_{2}\right) \approx 0.8776$
The maximum is $g \left(2 \pi\right) \approx 17.5663$