Let f(x)=x^(3)+6x^(2)+6x and let c be the number that satisfies the Mean Value Theorem for f on the interval [-6,0]. What is c ? Choose 1 answer: (A) -5 (B) -4 (C) -3 (D) -1

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Let  f(x)=x^(3)+6x^(2)+6x and let  c be the number that satisfies the Mean Value Theorem for  f on the interval  [-6,0]. What is  c ? Choose 1 answer: (A) -5 (B) -4 (C) -3 (D) -1

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Understand MVT: Understand the Mean Value Theorem (MVT) and what it states. The Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that f'(c) is equal to the average rate of change of the function over [a, b]. This can be written as: f'(c) = (f(b) - f(a)) / (b - a)Calculate Average Rate: Calculate the average rate of change of the function f(x) = x^3 + 6x^2 + 6x over the interval [-6, 0]. We need to find f(0) and f(-6) to use in the MVT formula. f(0) = 0^3 + 6*0^2 + 6*0 = 0 f(-6) = (-6)^3 + 6*(-6)^2 + 6*(-6) = -216 + 216 - 36 = -36 Now, we calculate the average rate of change: (f(0) - f(-6)) / (0 - (-6)) = (0 - (-36)) / 6 = 36 / 6 = 6Find Derivative: Find the derivative of the function f(x) = x^3 + 6x^2 + 6x. f'(x) = d/dx (x^3 + 6x^2 + 6x) = 3x^2 + 12x + 6Apply MVT: Apply the MVT by setting the derivative f'(c) equal to the average rate of change found in Step 2. We have f'(c) = 3c^2 + 12c + 6, and we want to set this equal to 6 (the average rate of change). 3c^2 + 12c + 6 = 6Solve Equation: Solve the equation 3c^2 + 12c + 6 = 6 for c. First, subtract 6 from both sides to set the equation to zero: 3c^2 + 12c + 6 - 6 = 0 3c^2 + 12c = 0 Now, factor out the common term 3c: 3c(c + 4) = 0Find Values of c: Find the values of c that satisfy the equation 3c(c + 4) = 0. We have two possible solutions for c: c = 0 or c + 4 = 0 Since c + 4 = 0 leads to c = -4, we have two potential values for c. However, c = 0 is not in the open interval (-6, 0), so we discard it. The value of c that satisfies the Mean Value Theorem for f on the interval [-6,0] is c = -4.

Let $f(x)=x^{3}+6 x^{2}+6 x$ and let $c$ be the number that satisfies the Mean Value Theorem for $f$ on the interval $[-6,0]$ . $\newline$ What is $c$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $-5$ $\newline$ (B) $-4$ $\newline$ (C) $-3$ $\newline$ (D) $-1$

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