Quadratic functions

1003206002

Level:

We are given three quadratic functions:

\begin{aligned} f_{1} (x) & = a x^{2} + 2 a x + a - 3, \\ f_{2} (x) & = a (x - 1)^{2} + 2, \\ f_{3} (x) & = a x^{2}, \end{aligned}

where

a \in (- \infty; 0)

. If possible, determine which of the given functions has the highest output value for

x = 0.5

f_{2}

f_{3}

f_{1}

Given information is insufficient to decide.

1003206001

Level:

We are given three quadratic functions:

\begin{aligned} f_{1} (x) & = - x^{2} - 2, \\ f_{2} (x) & = - x^{2} - 2 x - 4, \\ f_{3} (x) & = x^{2} + 2. \end{aligned}

Which of the given functions are increasing on the interval

(- 2; 0)

only

f_{1}

only

f_{2}

f_{1}

and

f_{2}

all three given functions

1003206202

Level:

Given

f (x) = - \frac{1}{2} x^{2} + x + \frac{3}{2}

, find all input values of

f

such that the output values of

f

are positive.

x \in (- 1; 3)

x \in (- \infty; - 1) \cup (3; + \infty)

x \in (- 3; 1)

x \in (- \infty; - 3) \cup (1; + \infty)

1003206201

Level:

Given

f (x) = 2 x^{2} - 6 x + 8

, find all input value(s) of

f

such that the output value of

f

5.5

x_{1} = \frac{5}{2}

x_{2} = \frac{1}{2}

x = 35.5

x_{1} = 13

x_{2} = 11

x_{1} = - \frac{5}{2}

x_{2} = - \frac{1}{2}

1003162309

Level:

Find all the values of the real parameter

p

such that

f (x) = p x^{2} - 4 p x + 4 p - 3

is a negative quadratic function for all

x \in R

p \in (- \infty; 0)

p = 0

p \in (0; \infty)

p \in (- 2; 2)

1003162308

Level:

Find all the values of the real parameter

p

such that

f (x) = (p - 2) x^{2} + p x + 2

has a maximum.

p \in (- \infty; 2)

p \in (- \infty; - 2)

p \in (2; + \infty)

p \in (- \infty; 0)

1003162307

Level:

Find all the values of the real parameter

p

such that

f (x) = 2 x^{2} + 3 p x + 2

has a minimum.

p \in (- \infty; \infty)

p \in (- \infty; 0) \cup (0; + \infty)

p = 0

p \in [0; \infty)

1003162306

Level:

Find all the values of the real parameter

p

such that

f (x) = 2 x^{2} + p x + p

is positive for all

x \in R

p \in (0; 8)

p \in (- \infty; 0) \cup (8; + \infty)

p \in (- \infty; 0)

p \in (0; \infty)

1003162305

Level:

Find all the values of the real parameter

p

such that

f (x) = 3 (x - 2)^{2} + p

is nonnegative for all

x \in R

p \in [0; \infty)

p \in (- \infty; 0)

p = 0

p \in (0; \infty)

1003162304

Level:

Find all the values of the real parameter

m

such that

f (x) = - x^{2} + 2 x m - m^{2} + 2

is increasing on

(- \infty; 0)

m \in [0; \infty)

m \in (- \infty; 0)

m \in (- \infty; 0]

m \in (- \infty; 2]

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