Understanding Functions and Matrices in Mathematics

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 fʹ(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 minimum 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 coefficient 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