Let f(x) be a twice differentiable function (i.e. f(x), f′(x) and f′′(x) are defined for all x), and let a ∈ R (a is a real number). If f′(a) = 0 and f′′(a) > 0, then a is a strict local minimum of f.
(a) Express the second sentence in propositional notation by identifying the parts of the statement (labeling them as p, q, etc.) and giving the form of the proposition.
(b) Express the converse, inverse, and contrapositive of the statement, using propositional logic, and in words. Simplify the expressions as much as possible (e.g. using DeMorgan’s laws).
(c) Negate the statement (i.e. if this theorem were not true, what would we be able to say?)
(d) Suppose that f′(a) = 0 and f′′(a) = 0. What (if anything) does the theorem allow you to conclude
about whether a is a strict local minimum? Explain.
for part a i believe f'(a)=0 is p and f"(a)>0 is q and "a is a strict local minimum of f" would then be r so you could say p^q->r
For part (a), you correctly identified p as "f'(a) = 0" and q as "f''(a) > 0", and denoted "a is a strict local minimum of f" as r. So, the statement can be expressed in propositional notation as p ∧ q → r.
Now let's move on to the other parts of the question:
(b) To express the converse, inverse, and contrapositive, we can use the following logical operators:
- Converse: q ∧ p → r
In words: If f''(a) > 0 and f'(a) = 0, then a is a strict local minimum of f.
- Inverse: ¬p ∧ ¬q → ¬r
In words: If f'(a) ≠ 0 or f''(a) ≤ 0, then a is not a strict local minimum of f.
- Contrapositive: ¬r → ¬q ∨ ¬p
In words: If a is not a strict local minimum of f, then either f''(a) ≤ 0 or f'(a) ≠ 0.
(c) To negate the statement, we can simply take its negation. The negation of p ∧ q → r is ¬(p ∧ q) ∨ r.
In words: If f'(a) = 0 and f''(a) > 0, then a is not a strict local minimum of f.
(d) If f'(a) = 0 and f''(a) = 0, the theorem does not allow us to conclude anything about whether a is a strict local minimum. The original statement requires f''(a) to be greater than 0 in order to guarantee that a is a strict local minimum. When f''(a) = 0, it means that the second derivative test is inconclusive, and we cannot determine the concavity of the function at point a. Therefore, we need the condition f''(a) > 0 to hold in order to ensure a is a strict local minimum.