Question

f(x)=|sin x+cos x+tan x+cot x+sec x+csc x|最小值

Answer 1

f(x)=|sin x+cos x+tan x+cot x+sec x+csc x|

=|(sin x+csc x)+(cos x+sec x)+(tan x+cot x)|

∵sin x+csc x≥2√(sin x·csc x)=2

或≤-2√(sin x·csc x)=-2,

同理cos x+sec x≥2或≤-2;

tan x+cot x≥2或≤-2;

∴(sin x+csc x)+(cos x+sec x)+(tan x+cot x)≥6或≤-6;

∴|(sin x+csc x)+(cos x+sec x)+(tan x+cot x)|≥6

即最小值是6