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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.

Problem Solving - Basic

         

If \(f(x)=\frac{ax+b}{x^2+1}\) has a local maximum of \(11\) at \(x=1\), what is the value of \(a+b\)?

If the polynomial \[f(x)=x^3+3ax^2+\frac{1}{4}bx+c\] has local extrema at \(x=-1\) and \(x=3,\) for real numbers \(a\), \(b\) and \(c\), what is the value of \(a+b\)?

A cubic function \(f(x)\) satisfies the following two conditions:

\(A.\) \(f(x)\) has an extreme value \( -4 \) at \( x = 1\ \) and

\(B.\) \(\displaystyle \lim_{x \to 0} \frac{f(x)}{x} = -4 \).

What is the value of \( f(4) \)?

If the sum of the local extrema of the function \( f(x) = x^3 - 3ax^2 + 9x + 27 \) is \( 0 \), what is the value of the real number \( a \)?

If the slope of the line tangent to the curve \[ y = ax^3 - bx^2 + cx \] at \( x = 2 \) is \( 15 \), and \( (1,\ 7) \) is the inflection point of the curve, what is the value of \( a + b + c \)?

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