# Understanding Functions and Matrices in Mathematics

Classified in Mathematics

Written at on English with a size of 4.21 KB.

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

FALSE: Domain is A; codomain is 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

#### 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

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’

TRUE

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

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

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

TRUE

True

True

True

TRUE

FALSE

FALSE