Finding all a such that \(\displaystyle{f{{\left({x}\right)}}}={\sin{{2}}}{x}\ \text{is}\

Ashleigh Mitchell

Ashleigh Mitchell

Answered question

2022-04-08

Finding all a such that f(x)=sin2x is 8(a+1)sinx+(4a2+8a14)x increasing and has no critical points

Answer & Explanation

Alan Bean

Alan Bean

Beginner2022-04-09Added 10 answers

f(x)=4(cos2x2(a+1)cosx+(a2+2a4))
We need to find a for which f(x) is increasing, which means f(x)>0.
Now, 4(cos2x2(a+1)cosx+(a2+2a4))>0
cos2x2(a+1)cosx+(a2+2a4)>0
(cos2x2(a+1)cosx+a2+2a+1)>5
((cosx(a+1))2>5
cosx(a+1)>5 or cosx(a+1)<5
a<cosx15
or a>cosx1+5
Hence, a<25 or a>5
Videoad3u

Videoad3u

Beginner2022-04-10Added 15 answers

Since I got the answer: To solve:
f(x)>0 i.e.
f(x)=4(;cos2x2(a+1)cosx+(a2+2a4);)>0
Completing the square: 4([cosx(a+1)]25)>0
Solving this, considering the critical cases according to what value cos x takes and final answer is:
a(,25)(5,)

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