Los números críticos de la función \( f(x)=x^{\frac{3}{5}}(4-x) \), son: a. \( x=-1 / 2 \) y \( x=3 / 2 \) b. 0 c. \( x=3 / 2 \) y \( x=0 \) d. \( x=1 / 2 \) y \( x=0 \) e. \( x=1 / 2 \) y \( x=-3 / 2 \)

The weekly revenue for a product is given by \( R(x)=290.4 x-0.033 x^{2} \), and the weekly cost is \( C(x)=12,000+145.2 x-0.066 x^{2}+0.00002 x^{3} \), where \( x \) is the number of units produced and sold. (a) How many units will give the maximum profit? (b) What is the maximum possible profit? (a) The number of units that will give the maximum profit is (Round to the nearest whole number as needed.) (b) The maximum possible profit is \( \$ \square \).

3) Sean las funciones: \[ f(x)=e^{x^{2} \cdot e^{2 x}} \quad g(x)=\left(\frac{x-3}{x^{2}+1}\right)^{2} \] Determine: a) \( f^{\prime}(x) \) b) \( g^{\prime}(x) \) c) Calcule el valor de \( \frac{g^{\prime}(0)-3 f(0)}{g(0)} \)

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the \( x \)-axis. \( y=\sqrt{x}, y=1, x=0 \) \( V=\square \) (Type an exact answer, using \( \pi \) as needed.)

The weekly revenue for a product is given by \( R(x)=324 x=0.036 x^{2} \), and the weekly cost is \( C(x)=9000+162 x-0.072 x^{2}+0.00003 x^{3} \), where \( x \) is the number of units produced and sold. (a) How many units will give the maximum profit? (b) What is the maximum possible profit? (a) The number of units that will give the maximum profit is (Round to the nearest whole number as needed.) (b) The maximum possible profit is \( \$ \square \).

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the \( x \)-axis. \( y=5 \sqrt{x}, y=5, x=0 \) \( V=\square \) (Type an exact answer, using \( \pi \) as needed.)