Let and . What can you say about ?

Question

Let  and . What can you say about ?

BlackTom AI · Accepted Answer

The question as provided is incomplete. It reads: 'Let  and . What can you say about ?' but the definitions or expressions for the function h(t) (and any other related variables) are missing, as are the specific answer choices. Without the actual function h(t) (or at least its formula), we cannot determine the location of local extrema or even the existence of local maxima/minima.

What we would normally do in a complete problem:
- Step 1: Obtain the explicit form of h(t). This lets us compute h'(t), the derivative, to locate critical points by solving h'(t) = 0 or where h'(t) is undefined.
- Step 2: Determine the nature of each critical point via the second derivative test (h''(t)) or by a first derivative sign analysis. A negative h''(t) at a critical point indicates a local maximum; a positive h''(t) indicates a local minimum; zero or inconclusive second derivative might require higher-order tests or a sign chart.
- Step 3: Consider endpoints if the domain is restricted, since the phrase 'local' maxima/minima pertains to interior points where the derivative vanishes.

If we take the claimed answer 'h(t) has only one local maximum which is at t = −2 and does not have a local minimum' at face value, this would be consistent with a parabola opening downward with its vertex (the only critical point) located at t = −2. However, many other function types can have a local maximum at t = −2 without being a simple downward-opening parabola. Without the actual h(t), this remains speculative.

In summary, to assess the statement, we need the explicit expression for h(t). Once provided, we would:
- compute h'(t), find all critical points
- evaluate h''(t) or use a monotonicity analysis around those points
- check whether there is exactly one critical point and whether it corresponds to a local maximum, and verify whether any other local extrema exist

Until the function is given, any claim about the number and location of local maxima/minima is not verifiable.

MATH-112-301-001 Unproctored Midcourse Exam 3 Practice Exam 1

Let and . What can you say about ?

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