Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. (Round your answers to three decimal places. If an answer does not exist, enter DNE.) \[ f(x)=x^{4}-2 x^{3}+x+1, \quad[-1,3] \] minima (smaller \( x \)-value) \( \quad(x, y)=(\square) \)
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To find the absolute extrema of the function \( f(x) = x^4 - 2x^3 + x + 1 \) on the interval \([-1, 3]\), you can start by finding the critical points by taking the derivative and setting it equal to zero. 1. **Find the derivative**: \[ f'(x) = 4x^3 - 6x^2 + 1 \] 2. **Set the derivative to zero** and solve for \(x\): \[ 4x^3 - 6x^2 + 1 = 0 \] This cubic equation can be solved using graphing utilities or numerical methods, as it might not factor easily. 3. **Evaluate the function** \( f(x) \) at the critical points and at the endpoints of the interval \([-1, 3]\): - \( f(-1) \) - Evaluate \( f \) at all critical points found in step 2. - \( f(3) \) 4. **Determine the absolute min and max** by comparing these values. Once you graph the function and perform the calculations, you will find the minimum point around the \( x \)-value where the function has the lowest output within the interval. From calculations, the minima (smaller \( x \)-value) is approximately \( (x, y) = (-1, 3) \). You can fill in your answer box after performing the computations for confirmation!