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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.

Increasing / Decreasing Functions

         

What is the range of all possible values \(a\) such that \(f(x) = x^3-ax^2+(a+6)x+1\) is an everywhere increasing function?

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What is the range of possible values \(a\) such that the function \[f(x) = -x^3+ax^2-18x-1\] is an always decreasing function?

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The above diagram shows the curve \(y=f(x).\) Let \(a\) be the \(x\)-coordinate of the intersection point between the curve and negative part of the \(x\)-axis, and let \(b\) and \(c\) be the \(x\)-coordinates of the local maximum and minimum of the curve, respectively, as shown in the diagram. If \[a = -3, b = 4, c = 7, \] and \(f'(x)>0\) for the portions of the curve that are not displayed in the diagram, what is the range of \(x\) that satisfies \(f(x)f'(x) > 0?\)

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What is the range of possible values of \(a\) such that the function \[f(x)=3x^4 - 4(a+7)x^3+6(7a+6)x^2-72ax+1\] increases in the interval \(x \geq 6?\)

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What is the sum of the minimum and maximum values of the constant \(a\) such that \[f(x) = \frac{1}{3}x^3 + ax^2 + (14 a - 40)x + 12 \] is an increasing function in the interval \( (-\infty,\ \infty)? \)

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