Calculus
# 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 $f(x) = 10{e}^x-10ex+8 ?$