check my answers please.
Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers.
f(x) = 2x3 + 10x2 - 8x - 5; between 1 and 2
-I got: f(1)=1 and f(2)=35;no
Determine the maximum possible number of turning points for the graph of the function.
f(x) = 8x^3 - 3x^2 + -8x - 22
-i got 2
f(x) = x^7 + 3x^8
i got 7
g(x) = - -1/x + 2
i got 0
I did not get an answer
To check your answers, let's go through each question and see if they are correct.
1. Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers.
f(x) = 2x^3 + 10x^2 - 8x - 5; between 1 and 2.
To apply the Intermediate Value Theorem, we need to evaluate the function at both ends of the interval and see if the signs change. Let's check:
f(1) = 2(1)^3 + 10(1)^2 - 8(1) - 5 = 2 + 10 - 8 - 5 = -1.
f(2) = 2(2)^3 + 10(2)^2 - 8(2) - 5 = 16 + 40 - 16 - 5 = 35.
You correctly calculated f(1) = -1 and f(2) = 35. Since the signs of f(1) and f(2) are different, this means that there is a real zero of the polynomial function f(x) = 2x^3 + 10x^2 - 8x - 5 between the integers 1 and 2.
Therefore, your answer of "no" is incorrect. There is indeed a real zero between 1 and 2.
2. Determine the maximum possible number of turning points for the graph of the function.
f(x) = 8x^3 - 3x^2 - 8x - 22.
To find the maximum possible number of turning points, we look at the degree of the polynomial function. In this case, the degree is 3. The maximum number of turning points for a polynomial function of degree n is n - 1.
So for f(x) = 8x^3 - 3x^2 - 8x - 22, the maximum possible number of turning points is 3 - 1 = 2.
Your answer of 2 is correct.
3. The polynomial f(x) = x^7 + 3x^8.
To find the degree of a polynomial, we identify the term with the highest exponent. In this case, the term with the highest exponent is 3x^8. Therefore, the degree of the polynomial is 8.
Your answer of 7 is incorrect. The correct answer is 8.
4. The function g(x) = -1/x + 2.
To determine if the function has any turning points, we need to find its derivative and see if it changes signs. The derivative of g(x) is g'(x) = 1/x^2.
Since the derivative g'(x) = 1/x^2 is always positive for all x ≠ 0, this means that there are no turning points for the function g(x) = -1/x + 2.
Your answer of 0 is correct.
So, in summary:
1. There is indeed a real zero between 1 and 2.
2. The maximum possible number of turning points for the graph of f(x) = 8x^3 - 3x^2 - 8x - 22 is 2.
3. The polynomial f(x) = x^7 + 3x^8 has a degree of 8.
4. The function g(x) = -1/x + 2 has 0 turning points.