Question

If x € R, find the minimum value of 3^x + 3^(1+x)

Answer 1

Hi friend

We know that AM >GM

For all x € R, We have

3^x + 3^(1-x)/2 > √3^x × 3^(1-x)

= 3^x + 3^(1-x)/2 = √3

= { 3^x + 3^(1+x) } > 2√3

hene,the minimum value of 3^x + 3^(1-x) is 2√3 for any x € R.

Answer 2

y = 3^x + 3^(1 + x) where x∈R
= 3^x + 3¹.3^x
= 3^x + 3.3^x
= 4.3^x
hence, y = 4.3^x
we know, exponential function is alwys positive for all real value .
y = 4.3^x
differentiate with respect to x
dy/dx = 4.3^x.log3 > 0 for all x∈R
hence, y is increasing function,
hence minnimum value of y is at x --> -∞
y = 4.3^(-∞) = 4.0 = 0
hence minnimum value of 3^x + 3^(1 +x) is 0
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