Consider the function f(x) = 5x^3 +5x^2 + 7, and let c be a number in the interval [0.1] . For what values of k is there a c in this interval such that ?
______≤k≤______
To find the values of k for which there exists a c in the interval [0,1] such that , we need to evaluate the function f(x) at the endpoints of the interval and observe the range of values spanned by the function.
First, let's find f(0) and f(1):
f(0) = 5(0)^3 + 5(0)^2 + 7 = 7
f(1) = 5(1)^3 + 5(1)^2 + 7 = 17
So, the range of the function f(x) over the interval [0,1] is from 7 to 17, inclusive:
7 ≤ k ≤ 17
Therefore, the values of k for which there exists a c in the interval [0,1] such that are given by:
7 ≤ k ≤ 17.