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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.

Concave / Convex Functions

         

What is the inflection point of the curve \[y=x^3-3x^2+7x+12?\]

What is the maximum value of real number \(a\) such that the curve \(y=x^2(\ln x-13)\) is concave down in the interval \((0,a)?\)

If \( f(x) = \ln (x^2 + 9) \), the interval on which the curve \( y = f(x) \) is concave up can be expressed as \( a < x < b \). What is the value of \( b - a \)?

If \( f(x) = x^4 - 32 x^3 + 288 x^2 + 8x - 15 \), 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 \(f(x) = x^4 - 40 x^3 + 504x^2 + 19x + 17\). Given that \(f(x)\) is concave in the domain \(\left[a,b\right]\), what is the value of \(a+b\)?

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