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Sagot :
Para resolver el problema, debemos seguir los pasos específicos para evaluar y comparar los valores de la función [tex]\( f(x) \)[/tex]. La función dada es [tex]\( f(x) = 2x^2 + 3x + 6 \)[/tex]. Vamos a calcular los valores de [tex]\( f(2) \)[/tex] y [tex]\( f(-3) \)[/tex], y luego encontrar la diferencia [tex]\( f(2) - f(-3) \)[/tex].
### Paso 1: Calcular [tex]\( f(2) \)[/tex]
Primero, sustituimos [tex]\( x = 2 \)[/tex] en la función [tex]\( f(x) \)[/tex]:
[tex]\[ f(2) = 2(2)^2 + 3(2) + 6 \][/tex]
Calculamos los términos uno a uno:
[tex]\[ (2)^2 = 4 \][/tex]
[tex]\[ 2 \cdot 4 = 8 \][/tex]
[tex]\[ 3(2) = 6 \][/tex]
Entonces:
[tex]\[ f(2) = 8 + 6 + 6 = 20 \][/tex]
### Paso 2: Calcular [tex]\( f(-3) \)[/tex]
Ahora, sustituimos [tex]\( x = -3 \)[/tex] en la función [tex]\( f(x) \)[/tex]:
[tex]\[ f(-3) = 2(-3)^2 + 3(-3) + 6 \][/tex]
Calculamos los términos uno a uno:
[tex]\[ (-3)^2 = 9 \][/tex]
[tex]\[ 2 \cdot 9 = 18 \][/tex]
[tex]\[ 3(-3) = -9 \][/tex]
Entonces:
[tex]\[ f(-3) = 18 - 9 + 6 = 15 \][/tex]
### Paso 3: Calcular [tex]\( f(2) - f(-3) \)[/tex]
Finalmente, encontramos la diferencia entre [tex]\( f(2) \)[/tex] y [tex]\( f(-3) \)[/tex]:
[tex]\[ f(2) - f(-3) = 20 - 15 = 5 \][/tex]
### Conclusión
El valor de [tex]\( f(2) - f(-3) \)[/tex] es [tex]\( 5 \)[/tex]. Por lo tanto, la respuesta correcta es:
C) 5
### Paso 1: Calcular [tex]\( f(2) \)[/tex]
Primero, sustituimos [tex]\( x = 2 \)[/tex] en la función [tex]\( f(x) \)[/tex]:
[tex]\[ f(2) = 2(2)^2 + 3(2) + 6 \][/tex]
Calculamos los términos uno a uno:
[tex]\[ (2)^2 = 4 \][/tex]
[tex]\[ 2 \cdot 4 = 8 \][/tex]
[tex]\[ 3(2) = 6 \][/tex]
Entonces:
[tex]\[ f(2) = 8 + 6 + 6 = 20 \][/tex]
### Paso 2: Calcular [tex]\( f(-3) \)[/tex]
Ahora, sustituimos [tex]\( x = -3 \)[/tex] en la función [tex]\( f(x) \)[/tex]:
[tex]\[ f(-3) = 2(-3)^2 + 3(-3) + 6 \][/tex]
Calculamos los términos uno a uno:
[tex]\[ (-3)^2 = 9 \][/tex]
[tex]\[ 2 \cdot 9 = 18 \][/tex]
[tex]\[ 3(-3) = -9 \][/tex]
Entonces:
[tex]\[ f(-3) = 18 - 9 + 6 = 15 \][/tex]
### Paso 3: Calcular [tex]\( f(2) - f(-3) \)[/tex]
Finalmente, encontramos la diferencia entre [tex]\( f(2) \)[/tex] y [tex]\( f(-3) \)[/tex]:
[tex]\[ f(2) - f(-3) = 20 - 15 = 5 \][/tex]
### Conclusión
El valor de [tex]\( f(2) - f(-3) \)[/tex] es [tex]\( 5 \)[/tex]. Por lo tanto, la respuesta correcta es:
C) 5
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