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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.

# 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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