Answer:
c = 0.858
Please note that my answer may be a bit off in the thousands place since I rounded early. Hope this helps!
Explanation:
1. The domain is all real numbers. Therefore, f(x) is continuous on [0,2]
2. The derivative is f'(x) = 2x + 3 - 8 cos(2x+3). Therefore, f(x) is differentiable on (0, 2).
3. Therefore, by MVT, there exists a point c on (0, 2) such that f'(c) = the slope of the secant line.
Let's find the slope of the secant line.
f(0) = -0.209
f(2) = 9.513
The slope is 4.861.
4.681 = 2c + 3 - 8 cos(2c + 3)
Use graphing calculator.