8. If the joint pdf of two random variables is given by \[ f(x, y)=\left\{\begin{array}{cc}k x y, & x^{2} \leqslant y \leqslant 1,0 \leqslant x \leqslant 1 \\ 0, & \text { otherwise }\end{array}\right. \] (1) Find the constant \( k \); (2) Find the probability \( P\{(X, Y) \in D\} \) where \( D=\left\{(x, y) \mid x^{2} \leqslant y \leqslant x, 0 \leqslant x \leqslant 1\right\} \)

Find the value of \( k \) that makes \( f(x) \) continuous at \( x=-5 \) \[ f(x)=\left\{\begin{array}{ll}\frac{-2 x^{2}-10 x}{x+5} & \text { if } x \neq \\ k & \text { if } x=\end{array}\right. \] \( k= \)

For the function below, (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or a minimum value and find that value; and (d) graph the function. \( f(x)=2 x^{2}+6 x+6 \) (a) The vertex is (Type an ordered pair, using integers or fractions:)

- \( \frac{5}{6}+\left[\left(\frac{7}{6}\right) \times \frac{5}{4}\right]= \)

sea \( f(x)=5 x+(4 x-1) \cdot \ln (2-x) \) Hallar la eciaciónde la recta tangente al gráfico de \( f(x) \) en \( x_{0}=1 \).

20. \( x^{2}-14 x+c \)