Find the x -coordinates of all relative maxima of f(x) i f(x)=-(3)/(4)x^4-3x^3+81x^2-37 Answer Attemptiout of 4 (4) Additional Solution No Solution z=

To find the x-coordinates of all relative maxima of the function $f}{4}x^{4}-3x^{3}+81x^{2}-37$, we need to find the critical points of the function and then determine which of these points correspond to relative maxima.<br /><br />Step 1: Find the derivative of the function $f(x)$.<br />$f'(x) = -3x^{3} - 9x^{2} + 162x$<br /><br />Step 2: Set the derivative equal to zero and solve for $x$ to find the critical points.<br />$-3x^{3} - 9x^{2} + 162x = 0$<br />$-3x(x^{2} + 3x - 54) = 0$<br />$-3x(x + 9)(x - 6) = 0$<br /><br />The critical points are $x = 0$, $x = -9$, and $x = 6$.<br /><br />Step 3: Use to determine the nature of each critical point.<br />Find the second derivative of $f(x)$.<br />$f''(x) = -9x^{2} - 18x + 162$<br /><br />Evaluate the second derivative at each critical point:<br />$f''(0) = 162 > 0$, so $x = 0$ is a relative minimum.<br />$f''(-9) = 0$, so $x = -9$ is an inflection point.<br />$f''(6) = 162 > 0$, so $x = 6$ is a relative minimum.<br /><br />Therefore, the x-coordinate of the relative maximum of the function $f(x)=-\frac{3}{4}x^{4}-3x^{3}+81x^{2}-37$ is $x = 6$.