▼❛t❤ ✶✸✷✿ ❉✐s❝✉ss✐♦♥ ❙❡ss✐♦♥✿ ❲❡❡❦ 3
     ❉✐r❡❝t✐♦♥s✿ ■♥ ❣r♦✉♣s ♦❢ 3✲4 st✉❞❡♥ts✱ ✇♦r❦ t❤❡ ♣r♦❜❧❡♠s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛❣❡✳ ❇❡❧♦✇✱ ❧✐st t❤❡ ♠❡♠❜❡rs ♦❢
     ②♦✉r ❣r♦✉♣ ❛♥❞ ②♦✉r ❛♥s✇❡rs t♦ t❤❡ s♣❡❝✐✜❡❞ q✉❡st✐♦♥s✳ ❚✉r♥ t❤✐s ♣❛♣❡r ✐♥ ❛t t❤❡ ❡♥❞ ♦❢ ❝❧❛ss✳ ❨♦✉ ❞♦ ♥♦t
     ♥❡❡❞ t♦ t✉r♥ ✐♥ t❤❡ q✉❡st✐♦♥ ♣❛❣❡ ♦r ②♦✉r ✇♦r❦✳
     ❆❞❞✐t✐♦♥❛❧ ■♥str✉❝t✐♦♥s✿ ❲❡✬❧❧ s♣❡♥❞ s♦♠❡ ♦❢ t❤❡ t✐♠❡ ♦♥ t❤✐s ✇♦r❦s❤❡❡t✱ ❛♥❞ s♦♠❡ ♦❢ t❤❡ t✐♠❡ r❡✈✐❡✇✐♥❣
     ❢♦r t❤❡ ❡①❛♠✳ ■t ✐s ♦❦❛② ✐❢ ②♦✉ ❞♦ ♥♦t ❝♦♠♣❧❡t❡❧② ✜♥✐s❤ ❛❧❧ ♦❢ t❤❡ ♣r♦❜❧❡♠s✳ ❆❧s♦✱ ❡❛❝❤ ❣r♦✉♣ ♠❡♠❜❡r s❤♦✉❧❞
     ✇♦r❦ t❤r♦✉❣❤ ❡❛❝❤ ♣r♦❜❧❡♠✱ ❛s s✐♠✐❧❛r ♣r♦❜❧❡♠s ♠❛② ❛♣♣❡❛r ♦♥ t❤❡ ❡①❛♠✳
     ❙❝♦r✐♥❣✿
                        ❈♦rr❡❝t ❛♥s✇❡rs ●r❛❞❡
                          ✵✕✶   ✵✪
                          ✷✕✸   ✽✵✪
                          ✹✕✺   ✶✵✵✪
     ●r♦✉♣ ▼❡♠❜❡rs✿
     ✺✳✸✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳
      ✭✶✮ ✭❛✮ F′(x) =
        ✭❜✮ G′(x) =
      ✭✷✮ F(x) =
      ✭✸✮ ✭❛✮ ❙t❛t❡ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✭s✮ ❛♥❞ ✇❤❡t❤❡r F ❤❛s ❛ ❧♦❝❛❧ ♠❛①✱ ❧♦❝❛❧ ♠✐♥✱ ♦r ♥❡✐t❤❡r ❛t ❡❛❝❤ ♦♥❡✿
        ✭❜✮ ❙t❛t❡ t❤❡ ✐♥✢❡❝t✐♦♥ ♣♦✐♥t✭s✮ ❛♥❞ ❤♦✇ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ F ❝❤❛♥❣❡s ❛t ❡❛❝❤ ♦♥❡✿
                                        ▼❛t❤ ✶✸✷ ❉✐s❝✉ss✐♦♥ ❙❡ss✐♦♥✿ ❲❡❡❦ ✷
          ✺✳✸✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳
              ✭✶✮ ❯s✐♥❣ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿
                                Z x2
                                  e   x+1
                   ✭❛✮ F(x) =         x−1dx
                                 2
                       ❙♦❧✉t✐♦♥✿ ▲❡t H(x) ❜❡ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ x+1✱ s♦ H′(x) = x+1✳ ❚❤❡♥
                                                                     x−1               x−1
                                                 Z ex2
                                                       x+1              x2
                                                  2    x−1dx=H(e )−H(2),
                                                 Z ex2                           
                                              d        x+1               2       2
                                                                      ′ x       x
                                              dx 2     x−1dx=H(e )· e               (2x)−0
                                                                       2            
                                                                      x
                                                                    e   +1          2
                                                                                   x
                                                                 =     2     · 2xe     .
                                                                      x
                                                                    e   −1
                   ✭❜✮ G(x) = Z x2 ln(x+3)dx.
                                 cosx
                       ❙♦❧✉t✐♦♥✿ ▲❡t K(x) ❜❡ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ln(x + 3)✱ s♦ K′(x) = ln(x + 3)✳ ❚❤❡♥
                                         Z 2
                                            x
                                                                2
                                           cosx ln(x + 3) = K(x ) −K(cosx),
                                         Z 2
                                       d    x
                                                              ′  2             ′
                                      dx cosxln(x+3) = K (x )·(2x)−K (cosx)·(−sinx)
                                                                 2
                                                         = ln(x +3)·(2x)+ln(cosx+3)sinx.
                                                                                                                     2         2
                                                                                                                              x
              ✭✷✮ ❯s✐♥❣ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❣✐✈❡ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ F(x) ♦❢ f(x) = sin (x) + e
                  s❛t✐s❢②✐♥❣ F(3) = 0. ❨♦✉r ❛♥s✇❡r ❝❛♥ ✐♥✈♦❧✈❡ ❛ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧✳
                  ❙♦❧✉t✐♦♥✿ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱
                                                d Z x              2                   2
                                                           2       t            2       x
                                               dx       sin (t) + e    dt = sin (x) +e .
                                                    a
                  ❚❤✉s✱ ❛♥② ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠              Z
                                                                x 2          t2
                                                     F(x) =        sin (t) + e    dt
                                                               a
                                              2         2
                  ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ sin (x) + ex ✳ ❍♦✇❡✈❡r✱ ✇❡ ❛❧s♦ ♥❡❡❞ ♦✉r ❛♥t✐❞❡r✐✈❛t✐✈❡ t♦ s❛t✐s❢② F(3) = 0✳
                  P❧✉❣❣✐♥❣ t❤❛t ✐♥✱ ✇❡ ✜♥❞ t❤❛t             Z                
                                                              3              2
                                                                    2       t
                                                       0 =       sin (t) + e    dt.
                                                             a                   R
                  ❖♥❡ ❡❛s② ✇❛② t♦ ❛❝❝♦♠♣❧✐s❤ t❤❛t ✐s ❜② s❡tt✐♥❣ a = 3✱ s✐♥❝❡       3 f(t)dt = 0✳ ❚❤✉s✱ ♦✉r ❛♥t✐❞❡r✐✈❛t✐✈❡ ✐s
                                                                                  3
                                                      Z x    2       t2        Z x
                                             F(x) =        sin (t) + e    dt =       f(t)dt .
                                                       3                           3
                               Z x 2
              ✭✸✮ ▲❡t F(x) = 0 (t −6t+8)dt.
                   ✭❛✮ ❋✐♥❞ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ F ✭✐✳❡✳ t❤❡ ♣♦✐♥ts ✇❤❡r❡ F′(x) = 0) ❛♥❞ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ❧♦❝❛❧
                       ♠✐♥✐♠❛ ♦r ❧♦❝❛❧ ♠❛①✐♠❛✳
                 ❙♦❧✉t✐♦♥✿ ❲❡ ❝♦✉❧❞ t❛❦❡ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ t♦ ❝♦♠♣✉t❡ F(x)✱ ❜✉t t❤❛t ✇♦✉❧❞ ❜❡ s✐❧❧② s✐♥❝❡ t❤❡ ♥❡①t
                 st❡♣ ✐s t♦ t❛❦❡ ❛ ❞❡r✐✈❛t✐✈❡✳
                 ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s t❡❧❧s ✉s t❤❛t
                            ′     d Z x 2             2
                           F (x) = dx 0 (t −6t+8)dt = x −6x+8=(x−2)(x−4).
                 ❚❤✉s✱ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ F(x) ♦❝❝✉r ✇❤❡♥
                                            0 = (x−2)(x−4),
                 t❤❛t ✐s✱ ✇❤❡♥ x = 2 ♦r x = 4✳
                 ■♥✈❡st✐❣❛t✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ F′(x) = (x−2)(x−4) ❢✉rt❤❡r✱ ✇❡ s❡❡ t❤❛t F′(x) > 0 ✇❤❡♥ x < 2 ♦r x >
                 4✱ ❛♥❞ F′(x) < 0 ✇❤❡♥ 2 < x < 4✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❢✉♥❝t✐♦♥ F(x) ✐s ✐♥❝r❡❛s✐♥❣ ✉♥t✐❧ ✇❡ ❣❡t t♦ x = 2✱ ❛t
                 ✇❤✐❝❤♣♦✐♥t✐tst❛rts ❞❡❝r❡❛s✐♥❣✱ s♦ F(x) ❤❛s ❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛t x = 2✳ ❆❢t❡r t❤❛t✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s
                 ❞❡❝r❡❛s✐♥❣ ✉♥t✐❧ x = 4✱ ❛t ✇❤✐❝❤ ♣♦✐♥t ✐t st❛rts ✐♥❝r❡❛s✐♥❣✱ s♦ F(x) ❤❛s ❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ❛t x = 4✳
              ✭❜✮ ❋✐♥❞ t❤❡ ♣♦✐♥ts ♦❢ ✐♥✢❡❝t✐♦♥ ♦❢ F ✭✐✳❡✳ t❤❡ ♣♦✐♥ts ✇❤❡r❡ F′′(x) = 0✮ ❛♥❞ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡
                 ❝♦♥❝❛✈✐t② ❝❤❛♥❣❡s ❢r♦♠ ✉♣ t♦ ❞♦✇♥ ♦r ❢r♦♠ ❞♦✇♥ t♦ ✉♣ ❛t ❡❛❝❤ ♦♥❡✳
                 ❙♦❧✉t✐♦♥✿ ❲❡ ❛❧r❡❛❞② ❝♦♠♣✉t❡❞ F′(x)✱ s♦✱ ✉s✐♥❣ t❤❡ ♣r♦❞✉❝t r✉❧❡✱ ✇❡ ❝♦♠♣✉t❡ t❤❛t
                          F′′(x) = d ((x−2)(x−4)) = 1·(x−4)+(x−2)·1=2x−6.
                                  dx
                 ❙♦❧✈✐♥❣✱ ✇❡ ✜♥❞ t❤❛t F′′(x) = 0 ✇❤❡♥ x = 3✳
                 ▲♦♦❦✐♥❣ ❛t t❤❡ ❡①♣r❡ss✐♦♥ 2x − 6✱ ✇❤❡♥ x < 3✱ ✇❡ s❡❡ t❤❛t F′′(x) < 0✱ s♦ F ✐s ❝♦♥❝❛✈❡ ❞♦✇♥✳
                 ❲❤❡♥ x > 3✱ ✇❡ s❡❡ t❤❛t F′′(x) > 0✱ s♦ F ✐s ❝♦♥❝❛✈❡ ✉♣✳ ❚❤✉s✱ ❛t t❤❡ ✐♥✢❡❝t✐♦♥ ♣♦✐♥t ❛t x = 3✱
                  F ❝❤❛♥❣❡s ❢r♦♠ ❜❡✐♥❣ ❝♦♥❝❛✈❡ ❞♦✇♥ t♦ ❜❡✐♥❣ ❝♦♥❝❛✈❡ ✉♣✳