Найдите количество целых чисел, принадлежащих множеству значений функции:
[tex]f(x)=16log_{ \frac{1}{6} } \frac{sinx+cosx+3 \sqrt{2} }{ \sqrt{2} } [/tex]
[tex]f(x)=16log_{ \frac{1}{6} } \frac{sinx+cosx+3 \sqrt{2} }{ \sqrt{2} } [/tex]
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Найдите количество целых чисел, принадлежащих множеству значений функции: f(x) =16Log(1/6) (sinx +cosx +3√2) /√2 .----------------------------------
f(x) =16Log(1/6) (sinx +cosx +3√2) /√2 =16Log(1/6) ( (sinx +cosx)/√2 +3) .
(sinx +cosx) / √2 =(1/√2) *sinx + (1/√2) *cosx) =
cos(π/4) *sinx + sin(π/4) *cosx = sin(π/4+x )
следовательно -1 ≤ (sinx +cosx) /√2 ≤ 1 ;
2 ≤ (sinx +cosx) /√2 +3 ≤ 4
т.к. 0 < 1/6 < 1 , то
Log(1/6) 2 ≥ Log(1/6) ( ( sinx +cosx)√2 +3 ) ≥ Log(1/6) 4 ;
16*Log(1/6) 2 ≥16* Log(1/6) ( ( sinx +cosx)√2 +3 ) ≥ 16* Log(1/6) 2² ;
32*Log(1/6) 2 ≤ f(x) ≤ 16* Log(1/6) 2 ;
-32*Log(6) 2 ≤ f(x) ≤ -16*Log(6) 2 ;
-32/(1+Log(2) 3) ≤ f(x) ≤ - 16 / (1+Log(2) 3 ) ;
{ -12 ; -11; -10 ; -9 ; -8 ; -7 }
ответ : 6 .
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