Giffen product graph

Classified in Mathematics

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1. F : A → R means that the codomain of f is A and its domain is R. 

FALSE: Domain is A; codomain is R
2. Points of form (x, f(x)), x ∈ A, belong to the graph of function f : B → A, 
where A ̸= B are non-empty subsets of R. 

 
3. A function f defined on R is called strictly increasing if f(x1) > f(x2) holds, 
whenever x1> x2. 

FALSE: A function, is strictly increasing if f(x0) < f(x1) whenever x0 < x1.
 
4. The derivative (a) of f at a is the slope of the tangent line to the graph of f 
at (a,f(a)). 

TRUE: y = f (x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f '(c) where f ' is the derivative of f.
 5. If fʹ(a) ≥ 0, then f is strictly increasing in a neighbourhood of a. 

FALSE: must be bigger than 0 6. If f : I → R has negative second derivative (everywhere in I, I is an open interval), then f is strictly concave in I. 

TRUE: If f : I → R is a continuous concave function on an interval I (the interval may be unbounded) and c ∈ I is a stationary point for f , then c is maximum point for f in c 
7. If f : D → R is a function, then a point c ∈ D is called a maximum point for f, if for all cʹ ∈ D it holds that f(c) ≥ f(cʹ). 

8. If f is a concave function defined on R (with fʹʹ(x) < 0 at any x ∈ R), then every stationary point for f is a local minimum of f. 

FALSE:  f’’(x) should be > 0.
9. Every (global) maximum point is a local maximum point. 


TRUE: the global maximum point is also a local maximum point
11. Every continuous function defined on a closed interval [a, b] possesses its mini- mum at a or at b or at a point c ∈ (a,b) where fʹ(c) = 0. 

FALSE: could be a minimum or maximum 12. A point a is called stationary point for f if fʹ(a) < 0. 

TRUE 13. If f and g are differentiable at a then the derivative of their product at a is (fg)ʹ(a) = fʹ(a)gʹ(a). 

FALSE: fg) 0 = f ‘g + fg’
 14. The chain rule for differentiation is (f ◦ g)ʹ(x) = fʹ(g(x))gʹ(x).
 TRUE
 
15. If f and g are differentiable at x and g(x) ̸= 0, then g (x) = (g(x))2 .  TRUE 16.   Every quadratic function (with non-zero coeffictient at x2) is concave. 

FALSE: if the coefficient is positive the function will be convex
17 .If f is an increasing differentiable function then its derivative is non-positive. 

FALSE: then its derivative must  be positive
18.If a function F is such that Fʹ(x) = f(x) for every x in the domain of f, then f (x)dx = F (x) + c, c ∈ R is a constant. 
 True 
 19.The definite integral of a continuous function (defined in a closed and bounded interval) is a function. 
FALSE: it will be the area of the given function
20. For a non-positive function g defined on an interval [a,b] its definite integral over I equals the area contained between the graph of g and x-axis. 
 True
21. For continuous functions f and g it holds f(x)g(x)dx = f(x)dx · g(x)dx. 

FALSE:  true for the sum of integrals
24. The product of two square matrices is defined if only if their orders are the same. 

TRUE
25. If orders of A, B, C are m × n, p × q, g × h, respectively, then their product D = ABC is defined if m = h or p = q, and D is of order m × h. 

True
26. The determinant of identity matrix of any order is 1. 

True
27. A square matrix A is called nonsingular if the determinant of A is not equal to 0. 
 True 
 28. The inverse of a square matrix A exists if, and only if, det(A) ̸= 0 

TRUE 
 29. To multiply a row of a matrix by any number is an elementary operation. 

FALSE
30. If Cramer’s rule applies to a system of n linear equations in n variables and the coefficient matrix is non-singular then the system has exactly one solution. 

FALSE
 

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