Conversiones para el trabajo ( 4186 Julio = 1 Kilocaloría)

regunta 3 Para qué valores de \( x \) la gráfica de \( f(x)=\frac{4}{3} x^{3}+\frac{31}{2} x^{2}-8 x+20 \) tiene unà recta tangente horizonta

Factor using the 'ac' method. (If the polynomial is prime, enter PRIME.) \( 60 n^{3}+28 n^{2}-144 n \)

53. \( f(x)=x+5, g(x)=2 x-3 \rightarrow\left(\frac{f}{g}\right)(x) \) 54. \( f(x)=2 x^{3}+1, g(x)=x^{2}+4 \rightarrow\left(\frac{f}{g}\right)(x) \) 55. \( f(x)=x^{3}-2, g(x)=x+1 \rightarrow\left(\frac{f}{g}\right)(x) \) 56. \( f(x)=x^{2}+3 x, g(x)=x^{2}+1 \rightarrow\left(\frac{f}{g}\right)(x) \) 57. \( f(x)=5 x^{2}-1, g(x)=x+4 \rightarrow\left(\frac{f}{g}\right)(x) \) 58. \( f(x)=3, g(x)=x^{2}-5 \rightarrow\left(\frac{f}{g}\right)(x) \) 59. \( f(x)=2 x+1, g(x)=\frac{1}{2} x-3 \rightarrow\left(\frac{f}{g}\right)(x) \) 60. \( f(x)=4 x^{3}, g(x)=2 x^{2}+1 \rightarrow\left(\frac{f}{g}\right)(x) \)

9. Suppose the joint pdf of \( (X, Y) \) is \[ f(x, y)=\left\{\begin{array}{cc}x^{2} y, & x^{2} \leqslant y \leqslant 1 \\ 0, & \text { otherwise }\end{array}\right. \] (1) Find the constant \( c \); (2) Find the marginal probability density functions of \( X \) and \( Y \).

Use the properties of logarithms to rewrite the following expression if possible. Assume that all variables represent positive real numbers. \( \log _{b} \frac{x y^{3}}{z^{5}} \)