Question

Wyznacz zbiór argumentów, dla których funkcja f przyjmuje wartości większe niż funkcja g.

a) f(x) = - x^2 + 4x +1 . g(x) = 3x^2 + 4x

b) f(x) = 3x^2 + x - 2 , g(x) = x^2 -2x

Answer 1

a)      f(x)>g(x)

 

-x²+4x+1>3x²+4x

 

-x²+4x+1-3x²-4x>0

 

-4x²+1>0      /*(-1)

 

4x²-1<0

 

(2x+1)(2x-1)<0

 

2x+1=0                                 2x-1=0

2x=-1                                       2x=1

x 1=-1/2                                   x=1/2

 

+++++++++++++                                        ++++++++++++++++++++++=

 

----------------------------o------------------o---------------------------------------->

                                    -1/2    - - - -       1/2      

 

Odp:x∈(-1/2;1/2)

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b)     3x²+x-2>x²-2x

 

     3x²+x-2-x²+2x>0

 

2x²+3x-2>0

 

Δ=9-4*2*(-2)=25

 

√Δ=5

 

x 1=(-3-5)/4=-2                         x 2=(-3+5)/4=1/2

 

 

+++++++++++                                                ++++++++++++++++++++++=

----------------------o-----------------------------o------------------------------------>

                           -2                                      1/2

                                 ------ - - - - - ---------

 

Odp x∈(-∞;-2)U(1/2;∞)

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