Si \( f(x) \) es continua en \( [a, b] \) y \( a<c<d<b \) y \( K=f(c)+f(d) \), demostran que existe \( x_{0} \) entre \( a \) y \( b \) tal que \( K=2 f\left(x_{0}\right) \).

Deriva 10 sigulente \( y=e^{-3 x^{2}+7 x} \)

Deriva 10 siguiente \( \left(3 x^{2}+6 x\right)\left(5 x^{2}+3 x\right) \)

A coolant tank is filled with coolant at a constant rate of \( 400 \sqrt[3]{t} \) gallons per hour. If the tank originally contained 200 gallons of coolant, how many gallons are in the tank after 8 hours?

Suppose that the profit (in dollars) from the production and sale of \( x \) units of a product is given by \[ P(x)=180 x-\frac{x^{2}}{100}-400 \text {. } \] In addition, suppose that for a certain month the number of units produced on day \( t \) of the month is \[ x=q(t)=1000+10 t \text {. } \] (a) Find \( (P \circ q)(t) \) to express the profit (in dollars) as a function of the day of the month. ( \( P \circ q)(t)=\square \) (b) Find the number of units produced, and the profit, on the nineteenth day of the month. units produced profit

Deriva lo siguiente \( y=\sqrt[3]{2 x^{2}+3 x} \)