(Originais Teachy 2024) - Question Hard of Mathematics

In an engineering project, an architect needs to calculate the amount of material needed to build a curved support, which follows the shape of a straight line function and has a cross-section that is a segment of a parabola. The formula for the height h(x) of the support in relation to the horizontal distance x is given by h(x) = 2x^2 - 3x + 5. To ensure structural stability, the architect needs to determine the highest point of the support and the ratio between the slopes of the tangent lines at the beginning and end of the support. For this, follow the steps: (1) Calculate the derivative of the function h(x) and find the roots of the derivative to determine the critical points that will correspond to the highest point. (2) Rationalize the expression of the slope of the tangent lines, which is a function of 1/√m, where m is the slope of the tangent line and can have different values. Considering that the derivative of the function h(x) is necessary for item 1 and the rationalization of denominators is necessary for item 2, answer the following items: (A) Calculate the derivative of h(x) and determine the coordinates of the critical point, which corresponds to the highest point of the support. (B) Let m be the slope of the tangent line at the beginning of the support (x=0) and n the slope at the end of the support (x=1). Rationalize the expressions of m and n, and calculate the ratio m/n of the slopes of the tangent lines. Remember to justify all steps of your calculation.