1003162308
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=(p-2) x^2+px+2 \) has a maximum.
\( p\in(-\infty;2) \)
\( p\in(-\infty;-2) \)
\( p\in(2;+\infty) \)
\( p\in(-\infty;0) \)
Quadratic functions
1003162307
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=2x^2+3px+2 \) has a minimum.
\( p\in(-\infty;\infty) \)
\( p\in(-\infty;0)\cup(0;+\infty) \)
\( p=0 \)
\( p\in[0;\infty) \)
1003162306
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=2x^2+px+p \) is positive for all \( x\in\mathbb{R} \).
\( p\in(0;8) \)
\( p\in(-\infty;0)\cup(8;+\infty) \)
\( p\in(-\infty;0) \)
\( p\in(0;\infty) \)
1003162305
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=3(x-2)^2+p \) is nonnegative for all \( x\in\mathbb{R} \).
\( p\in[0;\infty) \)
\( p\in(-\infty;0) \)
\( p=0 \)
\( p\in(0;\infty) \)
1003162304
Level:
C
Find all the values of the real parameter \( m \) such that \( f(x)=-x^2+2xm-m^2+2 \) is increasing on \( (-\infty;0) \).
\( m\in[0;\infty) \)
\( m\in(-\infty;0) \)
\( m\in(-\infty;0] \)
\( m\in(-\infty;2] \)
1003162303
Level:
C
Find all the values of the real parameter \( m \) such that \( f(x)=3(x+m)^2-2 \) is increasing on \( (0;\infty) \).
\( m\in[0;\infty) \)
\( m\in(-\infty;0) \)
\( m\in(-\infty;0] \)
\( m\in(-\infty;2] \)
1003162302
Level:
C
Find all the values of the real parameter \( m \) such that \( f(x)=-2(x-m)^2+3 \) is an even function.
\( m=0 \)
\( m=3 \)
\( m=-3 \)
\( m\in(-\infty;\infty) \)
1003162301
Level:
C
Find all the values of the real parameter \( a \) such that \( f(x)=ax^2-2 \) is decreasing on \( (0;\infty) \).
\( a\in(-\infty;0) \)
\( a\in(0;\infty) \)
\( a\in[2;+\infty) \)
\( a\in(-\infty;2] \)
1003108312
Level:
B
The graph of the function \( f \) is a parabola, vertex of which is \( [6;0] \) and \( f(2)= 8 \) is given. Find the function \( f \).
\( f(x)=\frac12(x-6)^2 \)
\( f(x)=-\frac12(x-6)^2 \)
\( f(x)=\frac12(x+6)^2 \)
\( f(x)=\frac12x^2+6 \)
1003108311
Level:
B
The quadratic function \( f \) has the minimum at \( x=-2 \) and its graph passes through the points \( [0;13] \), \( [-1; 4] \). Find the function \( f \).
\( f(x)=3(x+2)^2+1 \)
\( f(x)=-\frac59(x-2)^2+9 \)
\( f(x)=\frac59(x-2)^2+9 \)
\( f(x)=3(x+2)^2-1 \)