C

Step 1: f(x) = 4x³ - 5x² + x - 2[Write the function.]

Step 2: Since f(x) is continuous in [0, 1] and differentiable in (0, 1), Mean Value Theorem is applicable.

Step 3: f(0) = 4(0)³ - 5(0)² + 0 - 2 = - 2[Find f(0).]

Step 4: f(1) = 4(1)³ - 5(1)² + 1 - 2 = - 2[Find f(1).]

Step 5: f ′(x) = 12x² - 10x + 1[Find f ′(x).]

Step 6: f ′(c) = 12c² - 10c + 1[Find f ′(c).]

Step 7: By Mean Value Theorem, there exists c ∈ (0, 1) such that f ′(c) = f(1)-f(0)1-(0).

Step 8: 12c² - 10c + 1 = -2-(-2)1-0

Step 9: 12c² - 10c + 1 = 0

Step 10: c = 5-1312, 5+1312 ∈ (0, 1) [Use quadratic formula.]

Step 11: At x = 5+1312, 5-1312 in (0, 1), the function f(x) satisfies the Mean Value Theorem.