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)
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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) (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
Start with the given equation.
Plug in .
Cube to get .
Square to get .
Multiply and to get .
Multiply and to get .
Multiply and to get .
Combine like terms.
So (take note that the function value is negative)
-------------------------
Now let's evaluate the right endpoint 4
Start with the given equation.
Plug in .
Cube to get .
Square to get .
Multiply and to get .
Multiply and to get .
Multiply and to get .
Combine like terms.
So (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
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