How to use LDFLAGS in makefile

Your linker (ld) obviously doesn't like the order in which make arranges the GCC arguments so you'll have to change your Makefile a bit:

CC=gcc
CFLAGS=-Wall
LDFLAGS=-lm

.PHONY: all
all: client

.PHONY: clean
clean:
    $(RM) *~ *.o client

OBJECTS=client.o
client: $(OBJECTS)
    $(CC) $(CFLAGS) $(OBJECTS) -o client $(LDFLAGS)

In the line defining the client target change the order of $(LDFLAGS) as needed.

In more complicated build scenarios, it is common to break compilation into stages, with compilation and assembly happening first (output to object files), and linking object files into a final executable or library afterward--this prevents having to recompile all object files when their source files haven't changed. That's why including the linking flag -lm isn't working when you put it in CFLAGS (CFLAGS is used in the compilation stage).

The convention for libraries to be linked is to place them in either LOADLIBES or LDLIBS (GNU make includes both, but your mileage may vary):

LDLIBS=-lm

This should allow you to continue using the built-in rules rather than having to write your own linking rule. For other makes, there should be a flag to output built-in rules (for GNU make, this is -p). If your version of make does not have a built-in rule for linking (or if it does not have a placeholder for -l directives), you'll need to write your own:

client.o: client.c
    $(CC) $(CFLAGS) $(CPPFLAGS) $(TARGET_ARCH) -c -o $@ $<

client: client.o
    $(CC) $(LDFLAGS) $(TARGET_ARCH) $^ $(LOADLIBES) $(LDLIBS) -o $@