Questions: Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the polynomial function. f(x) = x^2 + 19x - x^5 The zero(s) of the function is/are at approximately -2.030, 0, 2.144. (Round to three decimal places as needed. Use a comma to separate answers as needed.) Find the relative maxima. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a relative maximum value of at x ≈ . (Round to three decimal places as needed.) B. The function does not have a relative maximum.

Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the polynomial function. f(x) = x^2 + 19x - x^5 The zero(s) of the function is/are at approximately -2.030, 0, 2.144. (Round to three decimal places as needed. Use a comma to separate answers as needed.) Find the relative maxima. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a relative maximum value of at x ≈ . (Round to three decimal places as needed.) B. The function does not have a relative maximum.
Transcript text: Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the polynomial function. \[ f(x)=x^{2}+19 x-x^{5} \] The zero(s) of the function is/are at approximately $-2.030,0,2.144$. (Round to three decimal places as needed. Use a comma to separate answers as needed.) Find the relative maxima. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a relative maximum value of $\square$ at $x \approx$ $\square$ . (Round to three decimal places as needed.) B. The function does not have a relative maximum.
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Solution

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Solution Steps

Step 1: Finding the Real Zeros

The real zeros of the polynomial function \( f(x) = x^2 + 19x - x^5 \) were estimated to be approximately \( 1.836 \).

Step 2: Finding the Critical Points

The critical points, where the derivative \( f'(x) = 2x + 19 - 5x^4 \) is equal to zero, were found to be approximately \( -1.344 \) and \( 1.447 \).

Step 3: Identifying the Relative Maximum

Evaluating the function at the critical points, we found that the function has a relative maximum value of approximately \( 23.243 \) at \( x \approx 1.447 \).

Final Answer

The real zeros of the function are approximately \( \boxed{1.836} \). The function has a relative maximum value of \( \boxed{23.243} \) at \( x \approx \boxed{1.447} \).

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