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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.

Second Derivative Test

         

What is the sum of all the local minima of \[f(x)=3x^4-8x^3-6x^2+24x+14?\]

Let \(y=ax+b\) be a line that intersects the curve \(f(x)=x^3-x\) at three points including its inflection point. Let \(c, d\) and \(e\) be the \(x\)-coordinates of the three intersection points with \(c<d<e.\) If \(f'(c)=5,\) what is \(f'(e)?\)

How many inflection points does the curve \(y=2xe^{-x^2}+7\) have?

What is the minimum value of integer \(a\) such that the curve \(y=7x^4-5x^3+ax^2\) has no inflection points?

Let \(a\) and \(b\) be the local maximum and local minimum of the function \[f(x)=-x^3+9x^2+13,\] respectively. What is the value of \(a-b?\)

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