Chapter 1.2, Problem 10E

a.

To determine

To find the logical connectives: if f has a relative minimum at $x_0$ and if f is differentiable at $x_0$ , then $f^{\prime }\left(x_0\right)=0$

b.

To determine

To find the logical connectives if n is prime, then $n=2$ or n is odd

c.

To determine

To find the logical connectives: R is symmetric and transitive whenever R is irreflexive

d.

To determine

To find the logical connectives: Bis square and not invertible whenever R is $\det B=0$

e.

To determine

To find the logical connectives: f has a critical point at $x_0$ iff $f^{\prime }\left(x_0\right)=0$ or $f^{\prime }\left(x_0\right)$ doesn’t exist

f.

To determine

To find the logical connectives: $2<n-6$ is a necessary condition for $2n<4$ or $n>4$

g.

To determine

To find the logical connectives: $6\ge n-3$ only if $n>4$ or $n>10$

h.

To determine

To find the logical connectives: x is Cauchy implies x is convergent

i.

To determine

To find the logical connectives: f is continuous at $x_0$ whenever $\underset{x\to x_0}{\lim }f\left(x\right)=f\left(x_0\right)$

j.

To determine

To find the logical connectives: If f is differentiable at $x_0$ and f is increasing at $x_0$ , then $f^{\prime }\left(x_0\right)>0$