高中指考◆數學甲題庫下載題庫

<先前分析>

 

f'(x)=12x²-22x+6

Let f'(x)=0 =>2(6x²-11x+3)=0

=>2(3x-1)(2x-3)=0 =>x=(1/3)、(3/2)

=>f(x) has critical point at x=(1/3) or (3/2)

f"(x)=24x-22

f"(1/3)=-14<0 =>f(x) has relative maximum value at x=(1/3)

f"(3/2)=14>0 =>f(x) has relative minimum value at x=(3/2)

=================================== 

(A) 函 數 f 的圖形在點 (1, -1)  的切線斜率為正 =>False

f'(x)=12x²-22x+6

=>f'(1)=12-22+6=18-22=-4

故，斜率為負

-------------------------------------------------------------------

(B) 函 數 f 的圖形與直線 y =1交於三點 =>False

f(x)在x=(1/3)  有局部極大值 (凹向下)=> 且 f(1/3)=25/27<1

故，僅交於一點

-------------------------------------------------------------------
(C) 函 數 f 的唯一相對極小值為=>True

f(x)的極值點在 x=1/3 & 3/2

f"(1/3)=-14<0 =>x=1/3處有相對極小值 f(1/3)=-9/4

f"(3/2)=14>0=>x=3/2處有相對極大值 f(3/2)=25/27

-------------------------------------------------------------------

(D)  =>True

Let f(x)=0 =>x=0、4/3、2

π>2 且 x>2 時 f'(x)遞增

故正確

-------------------------------------------------------------------
(E) =>False

cos(π)<cos(4π/7)<cos(π/2)

=>-1<cos(4π/7)<0

f(0)=0 且 x<0 時 f'(x)<0 (遞減)

故 f(cos(4π/7))<0

------------------------------------------------------

Answers：(C)(D)

f(x) 圖形 如下↓↓↓