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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.

Curve Sketching

Let $$P$$ and $$Q$$ be the two inflection points of the curve $$\displaystyle y=e^{-5x^2}.$$ What is the area of the triangle $$OPQ,$$ where $$O$$ is the origin?

If $$\displaystyle{f(x)=\frac{1}{x^2+9}},$$ the interval on which the curve $$y=f(x)$$ is concave down can be expressed as $$a<x<b.$$ What is the value of $$ab?$$

Let $$\displaystyle{f(x)=8+\frac{a}{x}+\frac{b}{x^2},}$$ where $$a$$ and $$b$$ are constants. If the point $$(-1,0)$$ is the inflection point of the curve $$y=f(x),$$ what is $$a+b?$$

If $$f(x) = x^{1/3}(x + 26)$$, the interval on which the curve $$y = f(x)$$ is concave down can be expressed as $$a < x < b$$. What is the value of $$a + b$$?

Let $$y=(ax^2+30)e^x$$ be the equation of a curve, where $$a$$ is a constant. If $$A$$ and $$B$$ are two distinct arbitrary points on the curve and the arc between the two points always lies below the line connecting the two points, what is the range of $$a?$$

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