In this task, it was necessary to define formulas for the function for which three features were given. The first of them was concerned about the decreasing of the chart at a given point. The second and third conditions were about the points of local minimum and maximum of the function: the so-called stationary points.
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First of all, we used the concept of a derivative for a function and its property to be equal to zero in stationary points. That is, the fact that the required function had minimum and maximum testifies to the fact that, in these values (-2, 2), the derivative function was zero. In addition, using the properties of the roots, the formula of the derivative was written as a multiplication of two brackets, which could be simplified: (x-2)(x+2)=x^2-4
Finally, the defining value was what was the minimum and what was the maximum. The fact is that the reduction and increase of the function at a specific point is due to signs: they are replaced periodically. Since the minimum function was at -2, so to the left of this value, the function was positive. From -2 to 2: negative. From 2 to infinity: positive again. Since the construction of an already existing result was not proof of this reasoning, we decided to change the signs of the full function to the opposite, in other words, we made an inversion relative to the X-axis. In this case, everything turned out to be correct.
Answer: -(1/3)*x^3 + 4x
This is a worthy and harmonious answer, which fully demonstrates the understanding of the material. I honestly liked that the author of the post used numerical, analytical methods, and showed his calculations in detail. Thus, he turned to the search of the function integral and showed me the rule he uses for this purpose. This is a surprisingly strict and academic style. In addition, the author noted the necessary points on the graph. Therefore, in general, this work is worthy of high appreciation because it reflects logic, fidelity of judgment, and mathematical literacy.