**Problem : **
Suppose Andrea decides to open a lemonade stand and her grandfather agrees to buy her
all the supplies she needs. Now she has to decide what price she will charge for the
lemonade. The more she charges per cup, the fewer cups people will buy.
Andrea (who happens to be a mathematician) figures that the number of cups she will sell
if she charges *x* cents per cup is approximately given by the function

c(x) = 1000e^{-x/5} |

What should Andrea charge to maximize her profits? How much will she make? Andrea's total profit is the number of cups sold times the number of cents received for each cup. This profit is given by the function

p(x) = 1000xe^{-x/5} |

with derivative

p'(x) | = | 1000xe^{-x/5} + e^{-x/5} | |

= | 1 - 1000e^{-x/5} |

Solving

**Problem : **
Find the minimum value of *f* (*x*) = - 2*x*^{3} +3*x*^{2} + 12*x* - 1 in the interval [1, 4].

f'(x) = - 6x^{2} + 6x + 12 = - 6(x + 1)(x - 2) |

Thus

f (1) | = | 12 | |

f (2) | = | 19 | |

f (4) | = | -31 |

so

**Problem : **
Does the function *g*(*x*) = 3 - 2*x* attain a minimum value on the interval (1, 3)?

g < g(x) |

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