Question

Prove that the function f(x) = tanx - 4x is strictly decreasing on (-pi/3, pi/3)

Answer 1

f(x) is strictly decreasing in interval (-π/3 , π/3) only when df(x)/dx <0 in this interval .

now , y = tanx -4x

differentiate wrt x

dy/dx =sec²x -4
=(secx+2)(secx-2)
now,
sec²x €(1 , 2)

so, dy/dx<0 so, f(x) is strictly decreasing function.