Question

In: Math

Let f(x) be differentiable on the interval [a,b]. Select all of the following statements which must...

Let f(x) be differentiable on the interval [a,b]. Select all of the following statements which must be true.

1. f(x) must attain its maximum value on the interval [a,b]

2. There is some number c in the interval (a,b) such that f′(c)= (f(b)−f(a))/ (b−a)

3. f(x) is integrable on [a,b].

4. f(x) is continuous on (a,b).

5. f(x) is a polynomial.

6. f(x) is an increasing function.

Solutions

Expert Solution


Related Solutions

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f...
Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f is non-decreasing on [a,b] if and only if f′(x) ≥ 0 for all x ∈ (a,b), while if f is non-increasing on [a,b] if and only if f′(x) ≤ 0 for all x ∈ (a, b). can you please prove this theorem? thank you!
f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f...
f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f rises strictly monotonously ⇒ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?) If f is reversable, f has no critical point. (TRUE or FALSE?) If a is a “minimizer” of f, then f ′(a)...
Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show...
Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show that there exists a point c ∈ (0, 2π) such that cos(c)f(c) + sin(c)f'(c) = 2 sin(c).
Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and...
Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and such that f(a)=f(b). Then there exists at least one point c on (a,b) such that f'(c)=0." Rolle's Theorem requires three conditions be satisified. (a) What are these three conditions? (b) Find three functions that satisfy exactly two of these three conditions, but for which the conclusion of Rolle's theorem does not follow, i.e., there is no point c in (a,b) such that f'(c)=0. Each...
Let f : Rn → R be a differentiable function. Suppose that a point x∗ is...
Let f : Rn → R be a differentiable function. Suppose that a point x∗ is a local minimum of f along every line passes through x∗; that is, the function g(α) = f(x^∗ + αd) is minimized at α = 0 for all d ∈ R^n. (i) Show that ∇f(x∗) = 0. (ii) Show by example that x^∗ neen not be a local minimum of f. Hint: Consider the function of two variables f(y, z) = (z − py^2)(z...
In the following statements f(theta; x) is the likelihood function. Select which of the following are...
In the following statements f(theta; x) is the likelihood function. Select which of the following are true statements about maximum likelihood estimates (MLE's). Hint: In general, maximizing a function f(x) is equivalent to minimizing -f(x). Group of answer choices MLE's are unbiased. MLE's minimize the negative log likelihood function, -log(f(theta; x)) MLE's are always easy to find. MLE's maximize the likelihood function f(theta; x). MLE's are not affected by outliers in the data.
1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c)...
1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c) = n and g(c) = d, write the equation of the tangent line at x = c using only the variables y, x, c, n, and d. You may use point-slope or slope-intercept but do not introduce more variables. 2) Let f(x) be a continuous, everywhere differentiable function. What kind information does f'(x) provide regarding f(x)? 3) Let f(x) be a continuous, everywhere differentiable...
Let f:(a, b) → R be a function and n∈N. Assume that f is n-times differentiable...
Let f:(a, b) → R be a function and n∈N. Assume that f is n-times differentiable and f^(n)(x) = 0 for all x∈(a,b). Show that f is a polynomial of degree at most n−1.
9. Let f (x) = x^3 − 10. Find all numbers c in the interval (-11,...
9. Let f (x) = x^3 − 10. Find all numbers c in the interval (-11, 11) for which the line tangent to the graph of f is parallel to the line joining (−11, f (−11)) and (11, f(11)). How many such numbers exist in the given interval? . 0 . 1 . 2 (correct) . 3 Enter points in increasing order (smallest first). Enter DNE in any empty answer blank. c = c = c = DNE (correct) 10....
Suppose that f is a differentiable function with derivative f" (x) = (x − 3)(x +...
Suppose that f is a differentiable function with derivative f" (x) = (x − 3)(x + 1)(x + 5). Determine the intervals of x for which the function of f is increasing and decreasing Explain why a positive value for f" (x) means the graph f(x) is increasing For f(x) = 2x2- − 3x2 − 12x + 21, find where f'(x) = 0, and the intervals on which the function increases and decreases Determine the values of a, b, and...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT