Question

2^sinx+2^cosx find the minimum value of this trignometry

Answer 1

Take the deritiivive
$ln(2)cos(x)2^{sin(x)}-ln(2)sin(x)2^{cos(x)}$
it is zero at pi/4 and it repeats at every pi
minimum at where slope goesfrom negative to positive
so at 5pi/4 and 13pi/4 so at every 2pi interval


evaluate origial at 5pi/4
$2^{sin(5pi/4)}+2^{cos(5pi/4)}$=
$(2^ \frac{- \sqrt{2} }{2} )+(2^ \frac{- \sqrt{2} }{2} )$=
$2(2^ \frac{- \sqrt{2} }{2} )$ or aprox 1.22509