SOLUTION: Use the intermediate valve theorem to determine if the polynomial f(x)=x^3-4x^+3x+2 has a zero Ci in the interval [-3,4]. a). No because f(-3)*f(4)<0 b). Yes because f(-3)*f(4)

Algebra ->  Graphs -> SOLUTION: Use the intermediate valve theorem to determine if the polynomial f(x)=x^3-4x^+3x+2 has a zero Ci in the interval [-3,4]. a). No because f(-3)*f(4)<0 b). Yes because f(-3)*f(4)      Log On


   

Question 148852: Use the intermediate valve theorem to determine if the polynomial f(x)=x^3-4x^+3x+2 has a zero Ci in the interval [-3,4].
a). No because f(-3)*f(4)<0
b). Yes because f(-3)*f(4)<0
c). Yes because f(-3)*f(4)>0
d). No because f(-3)*f(4)>0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To find if there is a zero in the the interval [-3,4], simply evaluate the endpoints. If the function values of the endpoints have opposite signs, then there is a zero in the interval.


Let's evaluate the left endpoint -3


f%28x%29=x%5E3-4x%5E2%2B3x%2B2 Start with the given equation.


f%28-3%29=%28-3%29%5E3-4%28-3%29%5E2%2B3%28-3%29%2B2 Plug in x=-3.


f%28-3%29=1%28-27%29-4%28-3%29%5E2%2B3%28-3%29%2B2 Cube -3 to get -27.


f%28-3%29=1%28-27%29-4%289%29%2B3%28-3%29%2B2 Square -3 to get 9.


f%28-3%29=-27-4%289%29%2B3%28-3%29%2B2 Multiply 1 and -27 to get -27.


f%28-3%29=-27-36%2B3%28-3%29%2B2 Multiply -4 and 9 to get -36.


f%28-3%29=-27-36-9%2B2 Multiply 3 and -3 to get -9.


f%28-3%29=-70 Combine like terms.


So f%28-3%29=-70 (take note that the function value is negative)


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Now let's evaluate the right endpoint 4




f%28x%29=x%5E3-4x%5E2%2B3x%2B2 Start with the given equation.


f%284%29=%284%29%5E3-4%284%29%5E2%2B3%284%29%2B2 Plug in x=4.


f%284%29=1%2864%29-4%284%29%5E2%2B3%284%29%2B2 Cube 4 to get 64.


f%284%29=1%2864%29-4%2816%29%2B3%284%29%2B2 Square 4 to get 16.


f%284%29=64-4%2816%29%2B3%284%29%2B2 Multiply 1 and 64 to get 64.


f%284%29=64-64%2B3%284%29%2B2 Multiply -4 and 16 to get -64.


f%284%29=64-64%2B12%2B2 Multiply 3 and 4 to get 12.


f%284%29=14 Combine like terms.


So f%284%29=14 (take note that the function value is positive)


Since the y-values of the endpoints change in sign, this means that there is a zero in the interval. If you multiply f(-3) and f(4) you'll get a negative number. So this means that the answer is b). Yes because f(-3)*f(4)<0