Question

The function f is defined by

X
f(x) = x² + 3x - 4 sin (2x + 3). Find all values of that
satisfy the conclusion of the Mean Value Theorem on the
interval [0, 2]. You may use a calculator and round to the
nearest thousandth.

Answer 1

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.