Analyzing function behavior

1003261905

Level: 
A
Find the local extrema of the function \[ f(x)=x-\ln⁡(1+x)\text{ .} \]
the local minimum at \( x=0 \)
the local minimum at \( x=0 \), the local maximum at \( x=-1 \)
the local maximum at \( x=0 \)
the local maximum at \( x=0 \), the local minimum at \( x=-1 \)
do not exist

1003261908

Level: 
A
Determine all the values of \( t \), \( t\in\mathbb{R} \), such that the function \[ f(x)=tx^3+(t+1)x^2-(t-2)x+3 \] has local extrema.
\( t\in\mathbb{R}\setminus\left\{\frac12\right\} \)
\( t\in\mathbb{R} \)
\( t\in\left(-\frac12;\frac12\right) \)
\( t\in\left(-\infty;-\frac12\right)\cup\left(\frac12;\infty\right) \)

1103163606

Level: 
A
The graph of \( f' \) is given in the figure. Find the local extrema of \( f \). (The function \( f' \) is the derivative of the function \( f \).)
local minimum at \( x=0 \), local maxima at \( x_1=-2 \) and \( x_2=3 \)
local minimum at \( x=-1 \), local maximum at \( x=2 \)
local minima at \( x_1=-2 \) and \( x_2=3 \), local maximum at \( x=0 \)
local minima at \( x_1=-2 \) and \( x_2=0 \), local maximum at \( x=3 \)
local minimum at \( x=-2 \), local maxima at \( x_1=0 \) and \( x_2=2 \)