Zadatak graf fun. 1 var.

Nacrtaj graf funkcije f(x)=(1/(1 + exp ((1/x)))). Izracunaj tocke infleksije.

Rjesenje

f = (1/(1 + Exp[(1/x)]))

Limit[f, x -> 0, 
      Direction -> -1]

Limit[f, x -> 0, Direction -> 1]

Limit[f, 
      x -> ∞]

Limit[f, x -> -∞]

fp = (∂_x) (1/(1 + Exp[(1/x)]))

f2[x0_] := (∂_x) fp  /.  x -> x0

\(\(\(
\)\(∂\_x\ fp\)\)\)

Simplify[%]

f2[-5. ]

f2[-1. ]

f2[-0.3452]

Plot[{f,  1/2}, {x, -11, 11}, AspectRatio -> 1]

Zadatak graf fun. 1 var.

Nacrtaj graf funkcije f(x) = ln(((e + x))^((1/x))) . Izracunaj tocke infleksije.

Rjesenje

\(\(\(
\)\(f = Log[\((E + x)\)^\((1/x)\)]\)\)\)

FindRoot[% == 0, {x, -1}]

\(\(\(
\)\(Limit[f, x -> \(-E\), 
      Direction -> \(-1\)]\)\)\)

\(\(\(
\)\(\(Limit[f, x -> 0, Direction -> 1]\)\(
\)
    \)\)\)

\(\(\(
\)\(Limit[f, x -> 0, 
      Direction -> \(-1\)]\)\)\)

\(\(\(
\)\(
\)\(Limit[f, 
      x -> ∞]\)\)\)

\(\(\(
\)\(
\)\(
\)\(Limit[f, 
      x -> ∞]\)\)\)

fp1 = (∂_x) Log[(E + x)^(1/x)]

(∂_x) fp1

FindRoot[% == 0, {x, -2}]

Plot[f, {x, -E, 10}, AspectRatio -> 1.5]