inf and sup for Darboux sum

Question

How can I find $inf$ and $sup$ for Darboux sum of function $$f(x)=5sin(2x)+2cos(5x)~?$$ I have line segment $[-\frac{3\pi}{2}; -\frac{\pi}{2}]$. Partition points are $-\frac{\pi}{2}, -\frac{2\pi}{3},-\frac{3\pi}{4}, -\frac{5\pi}{6}, -\frac{3\pi}{2}-\pi,-\frac{7\pi}{6}, -\frac{5\pi}{4}, -\frac{4\pi}{3}, -\frac{3\pi}{2}$. Main problem is that it is hard to find zeros of derivative.

Answer 1

Not a step-by-step calculation by hand but here's an exact solution:

$$ f'(x) = 10 \cos(2x) - 10 \sin(5x) = 10 (\cos(2x) - \cos(5x - \pi/2))$$

On the interval $[-3/2\pi, -1/2\pi]$, infimum and minimum coincide and supremum and maximum too. You can check that the extrema are at

$$\left\{-\frac{1}{6} (7 \pi ),-\frac{\pi }{2},-\frac{\pi }{2},-\frac{1}{14} (11 \pi ),-\frac{1}{14} (19 \pi ),-\frac{1}{14} (15 \pi )\right\}$$

with corresponding values $$\left\{-\frac{1}{2} \left(3 \sqrt{3}\right),0,0,7 \cos \left(\frac{\pi }{14}\right),-7 \cos \left(\frac{3 \pi }{14}\right),-7 \sin \left(\frac{\pi }{7}\right)\right\},$$

Hence we can read off that the maximum is at $$\left\{-\frac{11}{14} \pi ,7 \cos \left(\frac{\pi }{14}\right)\right\}$$ and the minimum at $$\left\{-\frac{19}{14} \pi ,-7 \cos \left(\frac{3 \pi }{14}\right)\right\}.$$