Find the x-coordinates of all relative minima of f(x). f(x)=-3x^(3)-36x^(2)+81 x-46 Answer: x=

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Find the  x-coordinates of all relative minima of  f(x).  f(x)=-3x^(3)-36x^(2)+81 x-46 Answer:  x=

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Find Critical Points: To find the relative minima of the function f(x), we need to find the critical points by taking the derivative of f(x) and setting it equal to zero. f(x) = -3x^3 - 36x^2 + 81x - 46 Let's find the first derivative f'(x). f'(x) = d/dx (-3x^3 - 36x^2 + 81x - 46) f'(x) = -9x^2 - 72x + 81Solve Quadratic Equation: Now, we need to find the values of x for which f'(x) = 0. -9x^2 - 72x + 81 = 0 To solve this quadratic equation, we can factor it or use the quadratic formula. Let's try to factor it first. -9(x^2 + 8x - 9) = 0 Now, we look for two numbers that multiply to -9 and add up to 8. The numbers are 9 and -1. -9(x + 9)(x - 1) = 0Identify Critical Points: We have two potential critical points from the factored equation: x + 9 = 0 or x - 1 = 0 Solving these, we get: x = -9 or x = 1 These are the critical points where the derivative is zero, but we need to determine which of these points are relative minima.Second Derivative Test: To determine if these critical points are relative minima, we can use the second derivative test. Let's find the second derivative f''(x). f''(x) = d/dx (-9x^2 - 72x + 81) f''(x) = -18x - 72Evaluate at x = -9: Now, we evaluate the second derivative at the critical points. For x = -9: f''(-9) = -18(-9) - 72 f''(-9) = 162 - 72 f''(-9) = 90 Since f''(-9) > 0, the function is concave up at x = -9, indicating a relative minimum.Evaluate at x = 1: For x = 1: f''(1) = -18(1) - 72 f''(1) = -18 - 72 f''(1) = -90 Since f''(1) < 0, the function is concave down at x = 1, indicating a relative maximum, not a minimum.

Find the $x$ -coordinates of all relative minima of $f(x)$ . $\newline$ $f(x)=-3 x^{3}-36 x^{2}+81 x-46$ $\newline$ Answer: $x=$

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Find the x x x-coordinates of all relative minima of f(x) f(x) f(x).\newlinef(x)=−3x3−36x2+81x−46 f(x)=-3 x^{3}-36 x^{2}+81 x-46 f(x)=−3x3−36x2+81x−46\newlineAnswer: x= x= x=

Find the $x$ -coordinates of all relative minima of $f(x)$ . $\newline$ $f(x)=-3 x^{3}-36 x^{2}+81 x-46$ $\newline$ Answer: $x=$