AP Calculus AB:: Analytical Applications

The Mean Value Theorem (MVT) states that if \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists at least one \(c\) in \((a,b)\) such that:

- **A)** \(f'(c) = 0\)
- **B)** \(f'(c) = \dfrac{f(b)-f(a)}{b-a}\)
- **C)** \(f(c) = \dfrac{f(b)-f(a)}{b-a}\)
- **D)** \(f(c) = 0\)

For \(f(x) = x^2\) on \([0,2]\), find the value(s) of \(c\) guaranteed by the Mean Value Theorem.

- **A)** \(c=0\)
- **B)** \(c=1\)
- **C)** \(c=2\)
- **D)** \(c=0.5\)

The Extreme Value Theorem guarantees that a function \(f\) has both an absolute maximum and absolute minimum on \([a,b]\) provided:

- **A)** \(f'(x) \ne 0\) anywhere on \([a,b]\)
- **B)** \(f\) is a polynomial
- **C)** \(f\) is continuous on the closed interval \([a,b]\)
- **D)** \(f\) is differentiable everywhere

The 'Candidates Test' for finding absolute extrema on \([a,b]\) requires evaluating \(f\) at which set of points?

- **A)** Every integer value between \(a\) and \(b\)
- **B)** Only the critical points
- **C)** The two endpoints \(a\) and \(b\), together with all critical points (where \(f'=0\) or \(f'\) is undefined) inside \((a,b)\)
- **D)** Only the endpoints \(a\) and \(b\)

A critical point of \(f\) is defined as a point \(c\) in the domain of \(f\) where:

- **A)** \(f(c) = 0\)
- **B)** \(f'(c) = 0\) or \(f'(c)\) is undefined
- **C)** \(f\) is discontinuous at \(c\)
- **D)** \(f''(c) = 0\)

For \(f(x) = x^3 - 3x^2\), find the critical point(s).

- **A)** \(x=0\) and \(x=2\)
- **B)** \(x=0\) and \(x=3\)
- **C)** \(x=0\) only
- **D)** \(x=3\) only

If \(f'(x) > 0\) for all \(x\) in an interval, what can be concluded about \(f\) on that interval?

- **A)** \(f\) is strictly decreasing
- **B)** \(f\) has a local maximum somewhere in the interval
- **C)** \(f\) is strictly increasing on that interval
- **D)** \(f\) is constant

The First Derivative Test determines whether a critical point \(c\) is a local max, local min, or neither, based on:

- **A)** Whether \(c\) is positive or negative
- **B)** The value of \(f(c)\) alone
- **C)** Whether \(f''(c)\) exists
- **D)** Whether \(f'(x)\) changes from positive to negative (local max), negative to positive (local min), or does not change sign (neither) as \(x\) passes through \(c\)