The minimum value of 3 ^ { \sin x } + 3 ^ { \cos x } is

3 ^ { 1 - { 1 } /{ \sqrt { 2 } } }

CAT Exam Model Paper 2 with solutions for free online practice

Question : 80 of 100

Marks: +1, -0

The minimum value of

3 ^ { \sin x } + 3 ^ { \cos x }

Solution:

(b) We know A.M ≥ G.M.

⇒ { 3 ^ { \sin x } + 3 ^ { \cos x } }/ { 2 } ≥√ { 3 ^ { \sin x } .3 ^ { \cos x } }

⇒3 ^ { \sin x } + 3 ^ { \cos x } ≥ 2 √{ 3 ^ { \sin x + \cos x } }

Now minimum value of sin x + cos x is at

x = { 5 \π }/ { 4 }

⇒sin x + cos x = -

√2

∴3 ^ { \sin x } + 3 ^ { \cos x }≥ 2 √{ 3 ^ { - √{ 2 } } } = 2.3 ^ { { -√ { 2 } }/ { 2 } } = 2 ( 3 ^ { { - 1 }/ { √ { 2 } } } )

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