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Task 1. Show that the function 
is decreasing everywhere.
Task 2. Find the extrema and intervals of monotonisity:
а) 
; б) 
; в) 
.
The answer: а) 
as 
as 
- decreases; 
- increases;
б) 
- decreases; 
- increases;
в) 
- increases, 
- decreases.
Task 3. Prove the inequality: 
.
Task 4. Find the greatest and the least values of the function: 
on
. The answer: 
and 
.
Task 5. Represent the number 
as two summands so that the sum of their cubs will be the least. The answer: 
and .
Task 6. What should be the height of the cone inscribed in the sphere or the radius 
if the cone’s lateral is the greatest? The answer: 
.
Task 7. Show that the graph of the function 
is convex everywhere.
Task 8. Find the points of inflection and the interval of convexity of the graph of the function 
.
The answer: 
– the point of inflection; 
– the interval of concavity; 
– the interval of convexity.
Task 9. Find the asymptotes of the function’s graphs: 1) 
;
2) 
. Answer: 1) 
; 2) 
.
Task 10. Investigate the functions and construct their draphs:
а) 
; б) 
; в) 
.
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