What is the minimum value of $(\sin x + \cos x + \csc (2x))^3$

Question

What is the minimum value of:

$$(\sin x + \cos x + \csc 2x )^3$$

let us consider that $0<x<\pi/2$

Use of differentiation is not allowed.!

Now we can use am gm to get the minimum value as 13.5...!!But the problem is that for this to holds $\sin x = \cos x = \csc(2x)$ which is not possible..

Is $\sec$ bounded from below? Are $\sin$ and $\cos$ bounded from above? — Guy, Apr 02 '14 at 05:38
i had solved it withh differentiation...but not by any other method.!! — maths lover, Apr 02 '14 at 05:39
@mathslover "bounded". I mean that do $\sin$ and $\cos$ have maximum values? Does $\sec$ have a minimum value? — Guy, Apr 02 '14 at 05:42
Is there a value of $x$ for which $sine$, $cosine$ or $secant$ not defined ? — Claude Leibovici, Apr 02 '14 at 08:01

score 3 · Answer 1 · answered Apr 02 '14 at 08:19

3

As $$lim_{x\to\frac\pi4^+}\sec 2x=-\infty$$ $\sin x+\cos x$ is bounded and $t\mapsto t^3$ is increasing, there is no minimum in $(0,\pi/2)$.

But in the interval $[0,\pi/4)$: $$\sin x+\cos x=\sqrt 2 \sin(x+\pi/4)$$ is increasing and $\sec 2x\ $ is also increasing, so their sum composed with the cube is increasing. Because this, the minimum is reached at $x=0$: $$(\sin 0+\cos x+\sec 0)^3=2^3=8.$$

answered Apr 02 '14 at 08:19

Martín-Blas Pérez Pinilla

41,546
4
46
89

can u tell at which value is this attained – maths lover Apr 02 '14 at 09:54
Point 0, value 8. – Martín-Blas Pérez Pinilla Apr 02 '14 at 09:55
see 0 is not allowed in domain – maths lover Apr 02 '14 at 11:24
Then there is not minimum. The infimum is 8. – Martín-Blas Pérez Pinilla Apr 02 '14 at 11:42
sry there was and editing error..it was cosec not sec..!! – maths lover Apr 05 '14 at 02:36

What is the minimum value of $(\sin x + \cos x + \csc (2x))^3$

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