24 On considère la fonction \( f \) définie pa \( \left\{\begin{array}{c}\text { pour } x<2, f(x)=\frac{x^{3}+x^{2}-5 x-2}{x-2} \\ \text { pour } x>2, f(x)=\sqrt{x^{7}-7} \\ f(2)=5\end{array}\right. \) 1. Justifie que pour tout nombre rée \( x^{3}+x^{2}-5 x-2=(x-2)\left(x^{2}+3 x+1\right. \) 2. Étudie la limite de \( f \) en 2.

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{7}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) \( \int_{2}^{7}[g(x)-f(x)] d x=\square \) 2 (Simplify your answer.)

The president of a company that manufactures toy cars thinks that the function \( P(c)=-2 c^{2}+14 c-20 \) represents the company's profit, where \( c \) is the number of cars produced, in thousands, and \( P(c) \) is the company's profit, in hundreds of thousands of dollars. Determine the maximum profit the company can earn.

Evaluate the integral. \( \int_{1}^{4}\left(x^{2}-1\right) d x=\square \) (Simplify your answer.)

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{2}^{7} 5 g(x) d x=\square \) (Simplify your answer.)

\( \int_{2}^{7} 5 g(x) d x=\square \) (Simplify your answer.)