What is the extreme f(x)=e^{0.5x}+256^{-0.5x} ?

The extreme f(x)=e^{0.5x}+256^{-0.5x} is Minimum(0.52341…,1.53343…)

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extreme f(x)=e^{0.5x}+256^{-0.5x}

extreme

Minimum (0.52341 \dots, 1.53343 \dots)

Solution steps

Find the critical points:

x = 0.52341 \dots

Domain of

e^{0.5 x} + 25 6^{- 0.5 x} : - \infty < x < \infty

Find intervals:

Decreasing

: - \infty < x < 0.52341 \dots,

Increasing

: 0.52341 \dots < x < \infty

Plug the extreme point

x = 0.52341 \dots

into

e^{0.5 x} + 25 6^{- 0.5 x} \Rightarrow y = 1.53343 \dots

Minimum (0.52341 \dots, 1.53343 \dots)

What is the extreme f(x)=e^{0.5x}+256^{-0.5x} ?
The extreme f(x)=e^{0.5x}+256^{-0.5x} is Minimum(0.52341…,1.53343…)