Example 6: Use the Graphing Strategy Use the graphing strategy and analyze the function \( f(x)=x^{4}+4 x^{3} \). State all the pertinent information and sketch the graph of \( f \).
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To analyze the function \( f(x) = x^4 + 4x^3 \), start by identifying its critical points through the first derivative. The first derivative is \( f'(x) = 4x^3 + 12x^2 \), which you can factor as \( 4x^2(x + 3) \). Setting this equal to zero gives critical points at \( x = 0 \) and \( x = -3 \). Evaluating the second derivative, \( f''(x) = 12x^2 + 24x \), helps determine the nature of these points. You'll find \( f''(0) > 0 \) implies a local minimum, while \( f''(-3) < 0 \) indicates a local maximum. Now for some fun! When sketching the graph, you can visualize a pointy upward-opening curve with a dip just after the local maximum and a gentle rise as x moves to the right. It has a y-intercept at \( (0, 0) \) since \( f(0) = 0 \). Peek into some interesting historical context: Polynomial functions like this one date back to ancient mathematics. The Greeks constructed geometrical proofs, but it wasn’t until the 17th century that mathematicians like Descartes began using algebraic methods for function analysis. They laid the groundwork for how we graph and interpret functions today! For some real-world application, polynomial functions describe various phenomena like projectile motion and profit models in business. This specific function modulates growth and can be used to model cubic growth patterns—perfect for analyzing how changes in inputs (like price in business) can result in non-linear profit growth, directly influencing strategies!