For the following function, find (a) the critical numbers, (b) the open intervals where the function is increasing, and (c) the open intervals where it is decreasing. f(x)=x^(4)+12x^(3)+28x^(2)+4

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For the following function, find (a) the critical numbers, (b) the open intervals where the function is increasing, and (c) the open intervals where it is decreasing.  f(x)=x^(4)+12x^(3)+28x^(2)+4

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Find Derivative of f(x): First, find the derivative of f(x) to locate critical numbers. f'(x) = 4x^(3) + 36x^(2) + 56x + 0Locate Critical Numbers: Set the derivative equal to zero to find critical points. 4x^(3) + 36x^(2) + 56x = 0Set Derivative Equal to Zero: Factor out the greatest common factor, which is 4x. 4x(x^(2) + 9x + 14) = 0Factor Out Common Factor: Now factor the quadratic equation. x(x + 7)(x + 2) = 0Factor Quadratic Equation: Solve for x to find the critical numbers. x = 0, x = -7, x = -2Solve for Critical Numbers: Use a sign chart or test values in the intervals to determine where f'(x) is positive or negative. Test intervals: (-∞, -7), (-7, -2), (-2, 0), (0, ∞)Determine Sign of f'(x): Choose test points: -8, -5, -1, 1 and plug into f'(x). f'(-8) = 4(-8)^(3) + 36(-8)^(2) + 56(-8) > 0 f'(-5) = 4(-5)^(3) + 36(-5)^(2) + 56(-5) < 0 f'(-1) = 4(-1)^(3) + 36(-1)^(2) + 56(-1) > 0 f'(1) = 4(1)^(3) + 36(1)^(2) + 56(1) > 0Test Intervals with Points: Determine the intervals of increase and decrease. f'(x) > 0 on (-∞, -7) and (-2, 0) and (0, ∞) so f(x) is increasing on these intervals. f'(x) < 0 on (-7, -2) so f(x) is decreasing on this interval.

For the following function, find (a) the critical numbers, (b) the open intervals where the function is increasing, and (c) the open intervals where it is decreasing. $\newline$ $f(x)=x^{4}+12 x^{3}+28 x^{2}+4$

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For the following function, find (a) the critical numbers, (b) the open intervals where the function is increasing, and (c) the open intervals where it is decreasing.\newlinef(x)=x4+12x3+28x2+4 f(x)=x^{4}+12 x^{3}+28 x^{2}+4 f(x)=x4+12x3+28x2+4

For the following function, find (a) the critical numbers, (b) the open intervals where the function is increasing, and (c) the open intervals where it is decreasing. $\newline$ $f(x)=x^{4}+12 x^{3}+28 x^{2}+4$