I have no idea totally lost haven't done any type of math in almost 30 years
f(x)= x^4-x^3-7x^2+x+6
all the real roots are bounded by -3 and 4. Explain why these are the upper and lower bounds
To find the upper and lower bounds for the real roots of the function f(x) = x^4 - x^3 - 7x^2 + x + 6, we can use the concept of bounding in mathematics.
First, let's understand the concept of bounding. If a real root exists within a certain range, we can determine upper and lower bounds for that root by testing the function at the endpoints of the range. If the sign of the function values changes when going from the lower bound to the upper bound, then there must be at least one real root within that interval, according to the Intermediate Value Theorem.
Now, let's calculate f(-3) and f(4) to determine the signs and apply the bounding concept:
1. Evaluate f(-3):
f(-3) = (-3)^4 - (-3)^3 - 7(-3)^2 + (-3) + 6
= 81 - (-27) - 7(9) - 3 + 6
= 81 + 27 - 63 - 3 + 6
= 48
2. Evaluate f(4):
f(4) = (4)^4 - (4)^3 - 7(4)^2 + (4) + 6
= 256 - 64 - 7(16) + 4 + 6
= 256 - 64 - 112 + 4 + 6
= 90
We observe that f(-3) = 48 and f(4) = 90. Since the values change sign from positive to negative as we move from the lower bound, -3, to the upper bound, 4, we can conclude that there is at least one real root between -3 and 4.
Therefore, -3 is the lower bound and 4 is the upper bound for the real roots of the given function f(x)= x^4-x^3-7x^2+x+6.