Al3

Page 1

№ 945

1) f (x ) = 3 x − x x , x > 0; 3 3 1. f ' (x ) = − ⋅ x, 2 x 2  3  1 − x  = 0  2 x 

f′(x) = 0;

1− x x

=0, x=1

+ 0

1

x = 1 – точка max, f(1) = 3 – 1 = 2;

2. f (x ) = 3 x − 2 x x , x > 0, f ' (x ) = 3 − 3 x ,

(

)

f′(x) = 0; 3 1 − x = 0 , x = 1 , x = 1, x = 1, точка max. 3. 1 ∈ (0; +∞), f(1) = 3 – 2 = 1.

+

1

№ 946

1) f(x) = e3x – 3x на (-1; 1), f′(x) = 3e3x – 3, f′(x) = 0, 3(e3x – 1) = 0, e3x = 1 – e0, x = 0, x = 0 – точка min, 0 ∈ (-1; 1), f(0) = e3⋅0 – 3 ⋅ 3 = 1; –

+

0

1 1 x −1 1 + , f′(x) = 0, =0, 2) f (x ) = + ln x на (0; 2), f ' (x ) = − 2 x x x2 x 1 x = 1, х = 1 – точка min, 1 ∈ (0;2), f (1) = + ln 1 = 1 . 1 +

– 1

№ 947

1) f (x ) = x 4 5 − x на (0; 5), x ⋅ (−1) 4(5 − x ) − x 20 − 5 x f ' (x ) = 4 5 − x + = = , 3 3 4 4 4 4 (5 − x ) 4 (5 − x ) 4 (5 − x )3

f′(x) = 0,

20 − 5 x 4 (5 − x )3 4

f (4 ) = 4 ⋅ 4 5 − 4 = 4 ;

74

= 0 , x = 4, x = 4 – точка max, 4 ∈ (0; 5),


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