Let g(x)=3x^(4)+8x^(3)+4. What is the absolute maximum value of g ? Choose 1 answer: (A) 4 (B) 36 (C) 116 (D) g has no maximum value

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Let  g(x)=3x^(4)+8x^(3)+4. What is the absolute maximum value of  g ? Choose 1 answer: (A) 4 (B) 36 (C) 116 (D)  g has no maximum value

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Understand Maximum Value Criteria: First, we need to understand that the absolute maximum value of a polynomial function occurs either at the endpoints of the domain or at critical points where the derivative is zero or undefined. Since g(x) is a polynomial function, it is continuous and differentiable everywhere, and its domain is all real numbers. Therefore, there are no endpoints to consider, and we need to find the critical points by taking the derivative of g(x) and setting it to zero.Find Derivative of g(x): Let's find the derivative of g(x): g'(x) = d/dx [3x^(4) + 8x^(3) + 4] Using the power rule for derivatives, we get: g'(x) = 12x^(3) + 24x^(2)Set Derivative Equal to Zero: Now, we set the derivative equal to zero to find the critical points: 12x^(3) + 24x^(2) = 0 We can factor out a common term of 12x^(2): 12x^(2)(x + 2) = 0Identify Critical Points: Setting each factor equal to zero gives us the critical points: 12x^(2) = 0  -->  x = 0 x + 2 = 0  -->  x = -2Evaluate Critical Points: We have two critical points: x = 0 and x = -2. To determine if these points are maxima, minima, or neither, we can use the second derivative test or simply plug these values into the original function to see which gives the highest value for g(x).Determine Absolute Maximum: Let's evaluate g(x) at the critical points: g(0) = 3(0)^(4) + 8(0)^(3) + 4 = 4 g(-2) = 3(-2)^(4) + 8(-2)^(3) + 4 = 3(16) - 8(8) + 4 = 48 - 64 + 4 = -16 + 4 = -12Comparing the values of g(x) at the critical points, we see that g(0) = 4 is the highest value. Since the function is a polynomial of even degree with a positive leading coefficient, it will go to positive infinity as x goes to positive or negative infinity. Therefore, the function has no absolute maximum value; it increases without bound.

Let $g(x)=3 x^{4}+8 x^{3}+4$ . $\newline$ What is the absolute maximum value of $g$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $36$ $\newline$ (C) $116$ $\newline$ (D) $g$ has no maximum value

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Let g(x)=3x4+8x3+4 g(x)=3 x^{4}+8 x^{3}+4 g(x)=3x4+8x3+4.\newlineWhat is the absolute maximum value of g g g ?\newlineChoose 111 answer:\newline(A) 444\newline(B) 363636\newline(C) 116116116\newline(D) g g g has no maximum value

Let $g(x)=3 x^{4}+8 x^{3}+4$ . $\newline$ What is the absolute maximum value of $g$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $36$ $\newline$ (C) $116$ $\newline$ (D) $g$ has no maximum value