- #1

Rishav

- 1

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Consider the function:

f(x)=x^x^x

At 0,f(x)=0 and d/dx=1

At 1,f(x)=1 and d/dx=1

At 2,f(x)=16 and d/dx=107.11

At 3,f(x)=7.625597485000*10^12 and d/dx=5.43324993100000*10^14

Why this? Check the slopes...

And: Slopes between 0-1

At 0.1, f(x)=0.16057 d/dx= 1.658

At 0.2, f(x)=0.31146 d/dx=1.350

At 0.3, f(x)=0.43215 d/dx=1.070

At 0.4, f(x)=0.52987 d/dx=0.890

At 0.5, f(x)=0.61255 d/dx=0.777

At 0.6, f(x)=0.68662 d/dx=0.716

At 0.7, f(x)=0.75740 d/dx=0.708

At 0.8, f(x)=0.82972 d/dx=0.747

At 0.9, f(x)=0.90862 d/dx=0.840

At 1.0, f(x)=1.00000 d/dx=1.000

The slopes between 0.2-0.7 are decreasing...why's that? Can anyone explain?

And one more function f(x)=x^x^x^x

At 0, d/dx=-infinity

At 0.1, d/dx=-1.528330000

At 0.2, d/dx=-0.37292

At 0.3 d/dx~0

At 0.4 d/dx=0.31333

At 0.5 d/dx=0.45030

At 0.6 d/dx=0.54327

At 0.7 d/dx=0.63323

At 0.8 d/dx=0.72329

At 0.9 d/dx=0.83695

At 1.0 d/dx=1.000

So can anyone say why does the slope decrease and then increase so rapidly? Hope my questions are not falling on deaf ears!