Question

and we check the value of \\(f\\) at the right endpoint of the interval \\(0,8\\):

\\(f(8) = 48\\).

the intermediate value theorem guarantees a solution to \\(f(x) = 37\\) in the interval \\((0,8)\\) because \\(f(0) \le 37 \le f(8)\\).

Explanation:

Step1: Identify the function type

The graph represents a quadratic function passing through the origin \((0,0)\) and vertex \((1,-1)\).

$$f(x) = (x-1)^2 - 1 = x^2 - 2x$$

Step2: Evaluate at the endpoint

Substitute \(x = 8\) into the function.

$$f(8) = 8^2 - 2(8) = 64 - 16 = 48$$

Step3: Re-evaluate based on standard curves

If the curve is a simple parabola \(f(x) = x^2\), then at \(x = 8\):

$$f(8) = 8^2 = 64$$

Answer:

\(f(8) = 64\)