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Extrema

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.

Local Extrema

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

$$f(x) = x^3 + 3ax^2 + \frac{1}{3}bx + c$$ 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, $$|3| = 3, |-2| = 2$$.

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

If $$f(x) = \frac{ax+b}{x^2+1}$$ has a local maximum of $$22$$ at $$x = 1,$$ what is the value of $$a + b?$$

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

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