errors when compiling c code with ffmpeg library

I've installed ffmpeg library in my new ubuntu 16.04 OS. When I tried to compile my c code, I got the following strange errors:

/home/widerstand/ffmpeg_build/lib/libavutil.a(hwcontext_vaapi.o): In function `vaapi_device_free':
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:861: undefined reference to `XCloseDisplay'
/home/widerstand/ffmpeg_build/lib/libavutil.a(hwcontext_vaapi.o): In function `vaapi_device_create':
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:891: undefined reference to `XOpenDisplay'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:896: undefined reference to `vaGetDisplay'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:903: undefined reference to `XDisplayName'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:893: undefined reference to `XDisplayName'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:917: undefined reference to `vaGetDisplayDRM'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:891: undefined reference to `XOpenDisplay'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:893: undefined reference to `XDisplayName'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:898: undefined reference to `XDisplayName'
/home/widerstand/ffmpeg_build/lib/libavutil.a(hwcontext_vdpau.o): In function `vdpau_device_create':
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:431: undefined reference to `XOpenDisplay'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:437: undefined reference to `XDisplayString'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:439: undefined reference to `XDefaultScreen'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:439: undefined reference to `vdp_device_create_x11'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:433: undefined reference to `XDisplayName'
/home/widerstand/ffmpeg_build/lib/libavutil.a(hwcontext_vdpau.o): In function `vdpau_device_free':
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:410: undefined reference to `XCloseDisplay'
collect2: error: ld returned 1 exit status
Makefile:30: recipe for target 'video_analysis' failed
make: *** [video_analysis] Error 1

I've reinstalled ffmpeg library multiple times using the following configurations:

PATH="$HOME/bin:$PATH" PKG_CONFIG_PATH="$HOME/ffmpeg_build/lib/pkgconfig" ./configure \
  --prefix="$HOME/ffmpeg_build" \
  --pkg-config-flags="--static" \
  --extra-cflags="-I$HOME/ffmpeg_build/include" \
  --extra-ldflags="-L$HOME/ffmpeg_build/lib" \
  --bindir="$HOME/bin" \
  --enable-gpl \
  --enable-libass \
  --enable-libfdk-aac \
  --enable-libfreetype \
  --enable-libmp3lame \
  --enable-libopus \
  --enable-libtheora \
  --enable-libvorbis \
  --enable-libvpx \
  --enable-libx264 \
  --enable-nonfree
PATH="$HOME/bin:$PATH" make

The installation process was successful each time. But when I tried to compile my own c code, I always encountered the same errors as mentioned above. I have no idea about that... Here is the makefile I used:

EDIT:

FFMPEG_LIBS=    libavdevice                        \
                libavformat                        \
                libavfilter                        \
                libavcodec                         \
                libswresample                      \
                libswscale                         \
                libavutil                          \

TARGET = video_analysis
LIBS = -lX11 -lm -lvdpau -lva
CC = gcc
CFLAGS += -O2 -g -O0
CFLAGS := $(shell pkg-config --cflags $(FFMPEG_LIBS)) $(CFLAGS)
LDLIBS := $(shell pkg-config --libs $(FFMPEG_LIBS)) $(LDLIBS)

.PHONY: default all clean

default: $(TARGET)
all: default

OBJECTS = $(patsubst %.c, %.o, $(wildcard *.c))
HEADERS = $(wildcard *.h)

%.o: %.c $(HEADERS)
    $(CC) $(CFLAGS) -c $< -o $@

.PRECIOUS: $(TARGET) $(OBJECTS)

$(TARGET): $(OBJECTS)
    $(CC) $(OBJECTS) $(LDLIBS) $(LIBS) -o $@

clean:
    -rm -f *.o
    -rm -f $(TARGET)

EDIT:

Afer swapping $(LIBS) and $(LDLIBS) to be $(LDLIBS) $(LIBS) at line 30, it looks much better, but the compiler still reports the following errors:

/root/ffmpeg_build/lib/libavutil.a(hwcontext_vaapi.o): In function `vaapi_device_create':
/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:896: undefined reference to `vaGetDisplay'
/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:917: undefined reference to `vaGetDisplayDRM'

I have no idea which library is still missing? It looks as if libva were still missing?... I got the answer by myself...Have a look at here: errors of 'vaGetDisplay' and `vaGetDisplayDRM'

You need to link with the libraries which provide those functions as well as FFmpeg.

The linker flags you need to add are (with their respective errors):

-lvdpau - libvdpau-dev

/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:439: undefined reference to `vdp_device_create_x11'

-lva - libva-dev

/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:896: undefined reference to `vaGetDisplay'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:917: undefined reference to `vaGetDisplayDRM'

-lX11 - libx11-dev

/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:861: undefined reference to `XCloseDisplay'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vaapi.c:898: undefined reference to `XDisplayName'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:431: undefined reference to `XOpenDisplay'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:437: undefined reference to `XDisplayString'
/home/widerstand/ffmpeg_sources/ffmpeg/libavutil/hwcontext_vdpau.c:439: undefined reference to `XDefaultScreen'

These flags need to be added after your -lav* linker flags.

Edit: OP added makefile

You can also use pkg-config for all the libraries.

FFMPEG_LIBS=    libavdevice                        \
                libavformat                        \
                libavfilter                        \
                libavcodec                         \
                libswresample                      \
                libswscale                         \
                libavutil

PKG_CFG_LIBS=   x11                                \
                vdpau                              \
                libva                              \
                $(FFMPEG_LIBS)

TARGET = video_analysis
LIBS = -lm
CC = gcc
CFLAGS += -O2 -g -O0
CFLAGS := $(shell pkg-config --cflags $(PKG_CFG_LIBS)) $(CFLAGS)
LDLIBS := $(shell pkg-config --libs $(PKG_CFG_LIBS)) $(LDLIBS)

.PHONY: default all clean

default: $(TARGET)
all: default

OBJECTS = $(patsubst %.c, %.o, $(wildcard *.c))
HEADERS = $(wildcard *.h)

%.o: %.c $(HEADERS)
    $(CC) $(CFLAGS) -c $< -o $@

.PRECIOUS: $(TARGET) $(OBJECTS)

$(TARGET): $(OBJECTS)
    $(CC) $(OBJECTS) $(LIBS) $(LDLIBS) -o $@

clean:
    -rm -f *.o
    -rm -f $(TARGET)