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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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