Given the function \( f(x)=\int_{1}^{x}\left(t^{4}-12 t^{3}-28 t^{2}\right) d t \), determine all \( x- \) values, if any, for which \( f \) has a relative minimum.

To determine the `x`-values for which the function \( f(x) \) has a relative minimum, you first need to find the derivative of the function. According to the Fundamental Theorem of Calculus, the derivative is given by: \[ f'(x) = t^{4} - 12 t^{3} - 28 t^{2} \bigg|_{t=x} = x^{4} - 12 x^{3} - 28 x^{2} \] Next, set the derivative equal to zero to find critical points: \[ x^{4} - 12 x^{3} - 28 x^{2} = 0 \] Factor out \( x^{2} \): \[ x^{2}(x^{2} - 12 x - 28) = 0 \] From here, \( x^{2} = 0 \) gives \( x = 0 \), and for \( x^{2} - 12 x - 28 = 0 \), you can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{12 \pm \sqrt{144 + 112}}{2} = \frac{12 \pm \sqrt{256}}{2} = \frac{12 \pm 16}{2} \] This results in: 1. \( x = 14 \) 2. \( x = -2 \) Now, to determine if these points are relative minima, you can use the second derivative test. Calculate \( f''(x) \): \[ f''(x) = 4x^{3} - 36x^{2} - 56x \] Evaluate \( f''(x) \) at the critical points: 1. At \( x = 0 \): \[ f''(0) = 4(0)^{3} - 36(0)^{2} - 56(0) = 0 \quad (\text{Inconclusive}) \] 2. At \( x = 14 \): \[ f''(14) = 4(14)^{3} - 36(14)^{2} - 56(14) \] Calculate term by term, and if \( f''(14) > 0 \), \( x = 14 \) is a local minimum. 3. At \( x = -2 \): \[ f''(-2) = 4(-2)^{3} - 36(-2)^{2} - 56(-2) \] Confirm \( f''(-2) < 0 \) shows \( x = -2 \) is a local maximum. Thus, the only critical point \( x = 14 \) would be where \( f(x) \) has a relative minimum. ### Conclusion: Therefore, the \( x \)-value for which \( f \) has a relative minimum is \( x = 14 \). --- History shows that calculus was formalized in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, who independently developed the concepts of differentiation and integration. Their principles laid the groundwork for understanding how functions behave and change over time, forming the backbone of classical mechanics. In the real world, finding relative minima is crucial in fields like economics for cost minimization, engineering for optimizing material usage, or even in machine learning for minimizing the error of predictive models. Understanding these points helps in making strategic decisions based on data analysis and optimization.