Question

Find the critical points of f(x)=(2^x)(sin(x) . I can only find -.96, but my online homework is telling me there are two more. Please explain

Answer 1

f' = 2^x cos(x) + 2^x ln(2) sin(x)
= 2^x ( cos(x) + ln(2) sin(x))

that doesnt appear to be undefined so it should amount to the zeros right?

Answer 2

Yes, in fact there are infinitely many solutions, as f'(x) = 0 when
tan x = -1/ln 2

Answer 3

yeah, those infinite ones have some extra jargon in them ... ;)

Answer 4

And we should expect an infinite number of critical points, as this will be an oscillating function where the amplitude increases in the positive x direction.

Answer 5

In physics, functions like this come up all the time.

Answer 6

how do i find the solutions within the interval [-2,6]?

Answer 7

by finding out when the derivative f'(x)= 0 between -2 and 6, and check the end points as well

Answer 8

since 2^x never equals zero, we only have to consider the rest of it:

cos(x) + ln(2) sin(x) = 0

ln(2) sin(x) = -cos(x)


-ln(2) tan(x) = 1

and

1 = -cot(x)/ln(2)

maybe....

Answer 9

how do i solve those for x? inverse cot(x)?