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 points

f (x) = e^{0.5 x} + 25 6^{- 0.5 x}

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)

f (x) = \frac{ln ( x )}{5 x ^{2}}

f (x, y) = 2 x^{2} + x + y^{2} - 2

2800 x - x^{3} + 3 x^{2} + 80 x - 500

f (x) = y = \frac{1}{3} x^{3} - 6 x + 1, - 1 \leq x \leq 6

y = 5 - (x + 1)^{2}

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…)