Analyzing function behavior

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.
\( t\in\mathbb{R}\setminus\left\{\frac12\right\} \)
\( t\in\mathbb{R} \)
\( t\in\left(-\frac12;\frac12\right) \)
\( t\in\left(-\infty;-\frac12\right)\cup\left(\frac12;\infty\right) \)

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

1003261904

Level: 
B
Given the function \[ f(x)=\sin ⁡x-3\cos⁡ x\text{ ,} \] determine the set of all \( x \), \( x\in\mathbb{R} \), such that \( f''(x)+f(x)=0 \).
\( \mathbb{R} \)
\( \emptyset \)
\( \{k\pi;\ k\in\mathbb{Z}\} \)
\( \left\{(2k+1)\frac{\pi}2;\ k\in\mathbb{Z} \right\} \)

1103163609

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 maximum at \( x=0 \)
local minimum at \( x=3 \), local maximum at \( x=0 \)
local minimum at \( x=1 \), local maximum at \( x=3 \)
local minimum at \( x=0 \), local maximum at \( x=3 \)
local minimum at \( x=0 \)

1103163608

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 \( x=3 \)
local minimum at \( x=2 \), local maximum at \( x=0 \)
local minimum at \( x=3 \), local maximum at \( x=0 \)
local minimum at \( x=0 \), local maximum at \( x=3 \)
local maximum at \( x=3 \)