Calculus

# Extrema Problem Solving

If $f(x)=\frac{ax+b}{x^2+1}$ has a local maximum of $11$ at $x=1$, what is the value of $a+b$?

If the polynomial $f(x)=x^3+3ax^2+\frac{1}{4}bx+c$ has local extrema at $x=-1$ and $x=3,$ for real numbers $a$, $b$ and $c$, what is the value of $a+b$?

A cubic function $f(x)$ satisfies the following two conditions:

$A.$ $f(x)$ has an extreme value $-4$ at $x = 1\$ and

$B.$ $\displaystyle \lim_{x \to 0} \frac{f(x)}{x} = -4$.

What is the value of $f(4)$?

If the sum of the local extrema of the function $f(x) = x^3 - 3ax^2 + 9x + 27$ is $0$, what is the value of the real number $a$?

If the slope of the line tangent to the curve $y = ax^3 - bx^2 + cx$ at $x = 2$ is $15$, and $(1,\ 7)$ is the inflection point of the curve, what is the value of $a + b + c$?

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