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How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.

What is the extreme value of the function \[ f(x) = x + 219 - \ln x ?\]

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\( f(x) = x^3 + 3ax^2 + \frac{1}{3}bx + c \) has local extrema at \( x = -1\) and \(x = 3 \), for real numbers \(a\), \(b\) and \(c\). What is the value of \( |a + b| \)?

**Details and assumptions**

The notation \( | \cdot | \) denotes the absolute value. The function is given by \[ |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} \] For example, \( |3| = 3, |-2| = 2 \).

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What is the value of the local maximum of the function \[ f(x) = -x^3 + 15x^2+17 ?\]

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What is the minimum value of \( f(x) = 10{e}^x-10ex+8 ?\)

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