Question 148852
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(x)=x^3-4x^2+3x+2}}} Start with the given equation.



{{{f(-3)=(-3)^3-4(-3)^2+3(-3)+2}}} Plug in {{{x=-3}}}.



{{{f(-3)=1(-27)-4(-3)^2+3(-3)+2}}} Cube {{{-3}}} to get {{{-27}}}.



{{{f(-3)=1(-27)-4(9)+3(-3)+2}}} Square {{{-3}}} to get {{{9}}}.



{{{f(-3)=-27-4(9)+3(-3)+2}}} Multiply {{{1}}} and {{{-27}}} to get {{{-27}}}.



{{{f(-3)=-27-36+3(-3)+2}}} Multiply {{{-4}}} and {{{9}}} to get {{{-36}}}.



{{{f(-3)=-27-36-9+2}}} Multiply {{{3}}} and {{{-3}}} to get {{{-9}}}.



{{{f(-3)=-70}}} Combine like terms.



So {{{f(-3)=-70}}} (take note that the function value is negative)



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Now let's evaluate the right endpoint 4





{{{f(x)=x^3-4x^2+3x+2}}} Start with the given equation.



{{{f(4)=(4)^3-4(4)^2+3(4)+2}}} Plug in {{{x=4}}}.



{{{f(4)=1(64)-4(4)^2+3(4)+2}}} Cube {{{4}}} to get {{{64}}}.



{{{f(4)=1(64)-4(16)+3(4)+2}}} Square {{{4}}} to get {{{16}}}.



{{{f(4)=64-4(16)+3(4)+2}}} Multiply {{{1}}} and {{{64}}} to get {{{64}}}.



{{{f(4)=64-64+3(4)+2}}} Multiply {{{-4}}} and {{{16}}} to get {{{-64}}}.



{{{f(4)=64-64+12+2}}} Multiply {{{3}}} and {{{4}}} to get {{{12}}}.



{{{f(4)=14}}} Combine like terms.



So {{{f(4)=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