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2. Higher order derivatives Find \( f^{\prime \prime \prime}(x) \) if \( f(x)=6 x^{5}-4 x^{2} \). 3. Graphing Where does the graph of \( y=\frac{x^{2}+9}{x} \) have a horizontal tangent line?

Ask by Gordon Knight. in Canada
Mar 14,2025

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**Problem 2. Higher Order Derivatives** Given \( f(x) = 6x^5 - 4x^2 \), the third derivative \( f^{\prime\prime\prime}(x) \) is: \[ f^{\prime\prime\prime}(x) = 360x^2. \] --- **Problem 3. Graphing** The function \( y = \frac{x^2 + 9}{x} \) has horizontal tangent lines at the points: \[ (3, 6) \quad \text{and} \quad (-3, -6). \]

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To find the third derivative \( f^{\prime \prime \prime}(x) \) of the function \( f(x) = 6x^5 - 4x^2 \), we start by taking the first derivative: \( f'(x) = 30x^4 - 8x \). The second derivative is \( f''(x) = 120x^3 - 8 \). Finally, the third derivative is \( f^{\prime \prime \prime}(x) = 360x^2 \). Voilà, there you have it! For the graph of \( y = \frac{x^2 + 9}{x} \), we first rewrite it as \( y = x + \frac{9}{x} \). To find where the tangent is horizontal, we take the derivative \( y' = 1 - \frac{9}{x^2} \) and set it equal to zero: \( 1 - \frac{9}{x^2} = 0 \). Solving gives us \( x^2 = 9 \), so \( x = 3 \) or \( x = -3 \). Thus, the horizontal tangents occur at these points on the graph!

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