Let $$f(x)=$$$$sin$$ x $$+$$$$cos$$ $$x+$$ $$tan$$ $$x+$$ arc $$sin$$ $$x+$$ arc $$cos$$ x $$+$$ arc $$tan$$ x . If M and m are maximum and minimum values of $$f(x)$$ , then their arithmetic mean is equal to

Question

Infinity Learn · Accepted Answer

Domain of f is [$$-1$$,$$1$$]$$fx=sinx+cosx+tanx+$$$$sin$$$$-1x+$$$$cos$$$$-1x+$$$$tan$$$$-1xfx=sinx+cosx+tanx+$$π$$2+$$$$tan$$$$-1xf$$'$$x=cosx-sinx+sec2x+0+11+x2$$ here,$$sec2$$≥$$1$$ and $$11+x2$$∈$$12$$,$$1$$ and  $$cosx-sinx$$ ∈$$-2$$,$$2hence$$ f'x>$$0$$⇒ f is increasingRange is [$$f(-1)$$,$$f(1)$$]∴$$fxmin=f-1=-sin1+cos1-tan1+$$π$$2-$$π$$4$$                   $$m=$$π$$4+cos1-sin1-tan1$$∴$$fxmax=f1=sin1+cos1+tan1+$$π$$2+$$π$$4$$                      $$M=3$$$$π$$$$4+cos1+sin1+tan1$$                       ⇒$$M+m2=$$π$$2+cos1$$

Let f(x)=sin x +cos x+ tan x+ arc sin x+ arc cos x + arc tan x . If M and m are maximum and minimum values of f(x) , then their arithmetic mean is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material