Question

prove if possible

(cos x)^n <= cos (nx) if 0 <= x <= pi/2 and 0 < n < 1

Answer 1

\[\cos^{n} x \le \cos(nx)\]

Answer 2

Not sure if I got this right, but have you used trial numbers?

Answer 3

to test the inequality ?

Answer 4

I have no clue.

Answer 5

Grade 12 Advanced Functions?

Answer 6

f'(x) - g'(x) > 0 implies f'(x) > g'(x)
right?

i think i found a method to figure it out

Answer 7

f(x) = cos (nx)
g(x) = cos^n(x)

f(x) - g(x) = cos (nx) - cos^n(x)
if x = 0 then
f(x) - g(x) = 0 - 0 =0

this is our starting point, the point at which both functions are equal
maybe if we show f'(x) > g'(x) on the interval then that is sufficient to prove the original inequality

Answer 8

I have no clue.

Answer 9

showing (cos(nx))' .>= (cos^n(x))' is sufficient after showing they begin at some point.

Answer 10

Ok so

f(x) = cos (nx)
f'(x) = -n sin(nx)

g(x) = cos^n(x)
g'(x) = -n cos^(n-1)(x) * sin x

f'(x) > g'(x) implies f'(x) - g'(x) = 0 so . . .

if we can show

-n sin(nx) - (-n cos^(n-1)(x) * sin x) =

\[-n \sin(nx) + n \cos^{n-1}(x) * \sin x > 0\]

is true on the interval.. hm

Answer 11

ok .. this becomes

\[n \left( -\sin(nx) + \cos^{p - 1}(x) * \sin x \right)\]

Answer 12

\[n \left( -\sin (nx) + \frac{ \sin x }{ \cos^{1 - p} (x) } \right)\]

reversing the order of the exponent's difference and putting cos in the denominator

Answer 13

ok since sin is increasing on the interval (0, pi/2), if we focus just on the two sin functions...

-sin(nx) + sin x

we know that n is fractional, so this portion of the expression will always be positive

also we know cos is positive on the interval (0, pi/2)

so we can conclude with certainty that

-sin(nx) + sin x / cos^(1 - p)(x) > 0

if we multiply this expression by positive n, we will show

f'(x) - g'(x) > 0

which is what we set out to prove. seems ok to me

Answer 14

It looks quite ingenious to me!
I think p should be n, btw...
You have done a good job (yourself, that is...)!

Answer 15

youre right, the question uses p for the exponent but i'm used to n to denote an exponent . or k

Answer 16

wow i should seriously review what i type before i post it. lets just say p = n