Irrerbthist6n

2021-12-26

Finding all a such that $f\left(x\right)=\mathrm{sin}2x-8(a+1)\mathrm{sin}x+(4{a}^{2}+8a-14)x$ is increasing and has no critical points

Obviously, the first thing I did was to find the derivative of this function and simplify it a bit and I got:

${f}^{\prime}\left(x\right)=4({\mathrm{cos}}^{2}x-2(a+1)\mathrm{cos}x+({a}^{2}+2a-4))$

But now how do I proceed further, had it been a simple quadratic in x.

Obviously, the first thing I did was to find the derivative of this function and simplify it a bit and I got:

But now how do I proceed further, had it been a simple quadratic in x.

Cheryl King

Beginner2021-12-27Added 36 answers

We need to find a for which f(x) is increasing, which means f'(x)>0.

Now,

or

Hence,

kalupunangh

Beginner2021-12-28Added 29 answers

Since I got the answer: To solve:

Completing the square:

Solving this, considering the critical cases according to what value cosx takes and final answer is:

karton

Expert2022-01-08Added 613 answers

A preliminary exploration indicates that the answer is

Here's the full solution. We want:

Let

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