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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.
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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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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