Question 1047332
Well, I am sending this commentary to "ikleyn"--

Of course, it will be "SLIGHTLY SHIFTED".  

One MISTAKE you made was that you plotted {{{z = w^7 + (1-w)^4}}} as if x was the independent variable in the function, and over the SAME COORDINATE SYSTEM!  It was NOT.  

Remember, I let {{{w = cos^2(x)}}}.  
So naturally, the lowest points of the two graphs will NOT jive.  BUT, what is  guaranteed is that they will have the SAME LOWEST level.

Another MISTAKE you made was that, z is a composite function, and as such, its domain, as a function of w, is only {{{0 <= w <= 1}}}.  Meaning, your green-colored graph is WRONG.  
The range of z is [{{{( (-root(3,4) + sqrt( root(3,16)+4root(3,28)))/(2*root(3,7)) )^7 + (1- (-root(3,4) + sqrt( root(3,16)+4root(3,28)))/(2*root(3,7)) )^4}}},1], or[0.05310578625,1] (approximately).

In fact, I can even predict/estimate the amount of "shift" that you're talking about:

{{{cos^-1(sqrt(0.5860760201)) - 0.5860760201 = 0.1128151882}}}.


Naughty, naughty, not so smarty....


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{{{sin^8(x)+cos^14(x)}}}

= {{{(sin^2(x))^4+(cos^2(x))^7}}}

= {{{(1-cos^2(x))^4+(cos^2(x))^7}}}

Now let {{{w = cos^2(x)}}}.  ===> {{{ 0<= w <= 1}}}, and 

===>  {{{(1-cos^2(x))^4+(cos^2(x))^7 = w^7 + (1-w)^4}}}.

Let {{{z = w^7 + (1-w)^4}}}.

===> {{{dz/dw = 7w^6 - 4(1-w)^3}}}.  Letting this equal to zero, we get

{{{7w^6 = 4(1-w)^3}}}.

===> {{{root(3, 7)*w^2 = root(3,4)*(1-w)}}}, after taking cube root of both sides.

===> {{{root(3, 7)*w^2 + root(3,4)*w - root(3,4) = 0)}}}.

===> {{{w = (-root(3,4) +- sqrt( root(3,16)+4root(3,28)))/(2*root(3,7)) }}} 

= {{{w = (-root(3,4) + sqrt( root(3,16)+4root(3,28)))/(2*root(3,7)) }}}, since {{{ 0 <= w <= 1}}}.

&#8776; 0.5860760201.


Now to the left of 0.5860760201, z' < 0, so z is decreasing on [0, 0.5860760201).

To the right of 0.5860760201, z' > 0, so z is increasing on (0.5860760201,1].


Hence there is a local minimum at w = 0.5860760201.

We only need to check the other critical points where the derivative doesn't exist, namely at w = 0 and w = 1, 
to determine the other possible extrema.

At w = 0, z = 1.
At w = 1, z = 1.


Therefore the maximum of {{{sin^8(x)+cos^14(x)}}} is {{{highlight(1)}}}, while the minimum is 

{{{0.5860760201^7 + (1-0.5860760201)^4 = highlight(0.05310578625)}}}, approximately.