(a) Let f [a, b] → R be a function such that f is continuous on [a, b] and differentiable on Ja, b[. Assume, furthermore, that f(a) ≥ 0 and that f'(x) > 0 for all x Ela, b[. Use the mean value theorem to prove that f(x) > 0 for all x ]a, b]. No other method will be accepted! (b) Prove that sinx < x for all x > 0. You may use, without proof, the fact that sinx is differentiable on R with (sin x)' = cosx. Hint: Prove first that sin x < x for all x €]0,].

Q: Suppose f : A → B is a function, X ⊂ A and Y ⊂ B. What does it mean to say: (a) z is in the range of…

A:

Q: 1. y = c1e-2x + cze²x + cze2x 2. y = c1e* + cze3x

A:

Q: Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). Which of the…

A: Given: Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). To Find:…

Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Let f:A-B and X,Y be subsets of B. Then f-¹(XnY) = f'(X) nf-¹(Y). True False

A:

Q: Show that the function f (z) = Z is continuous at the point z=0 but not differentiable at z=0.

A:

Q: Show that if f(x) is continuous in(a, b),then there exists a point in(a, b) such that…

A:

Q: Let f (x) and g(x) be differentiable functions in (a, continuous at a and b, and g(x) #0 in [a, b].…

A:

Q: tion f defined as xy Ax, y)=- when (x, y)#(0, 0) when (x, y)=(0, 0) =0, is continuous at the origin.

A:

Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

A:

Q: Suppose f is defined on [a, b] and g is defined on [b, c] with f(b) g(b). Then define { f(x) if a <…

A:

Q: The question from the book calculus with concept in calculus Suppose that /has an inverse and is…

A:

Q: If f (s Continuous then on -fex) True or felse?

A:

Q: Let /: X →Y be a function, A, BC X and C, DCY. Show that: 1. f(AUB) = f(A) uf(B) 2. If AC B, then…

A:

Q: Let A be a skew-symmetric n X n matrix. (a) Prove that xT Ax= 0 for all x in Rn. (b) Prove that I+ A…

A:

Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

A: The given functions f,g and h are continous on [a,b] and differntiable on (a,b). So intermediate…

Q: Let f(x) — sin(nx) n³ n=1 Show that f is differentiable on R and that f' is continuous on R.

A: Within its interval of convergence, the derivative of a power series is the sum of derivatives of…

Q: Suppose that f is differentiable on [a, b]. Prove that if f'(a)f'(b)< 0 then there exists a point c…

A:

Q: 7. Show that if f is differentiable on (a, b), and f'(c) > 0, where ce (a, b) and if f' is…

A:

Q: 5. Assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). Prove there are x, y = [0, 1] with |x -…

A: We have to assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). and prove that there are x, y ∈…

Q: If f is a continuous function on [a, b], ƒ is differentiable on (a, b), and if both f(a) and f(b)…

A: This question is based on Rolle's theorem which is explained in step 2.

Q: Let f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not…

A:

Q: be de by 8 <x3 + y3 f(x, y) = x2 + y2 if (x, y) = (0, 0) if (x, y) = (0, 0). %3D (a) Is f continuous…

A: Note- Since you have posted a question with multiple sub-parts, we will solve first three sub-parts…

Q: Assume that f: R→ R and g: R → R are infinitely differentiable functions. Prove the following…

A: We have given that ,  are both infinitely differentiable.We need to prove that , We will use…

Q: If f', and y are continuous on [a, b] and derivable on Ja, b[, then show that there is a value c…

A:

Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

A: What is Continuous Function: A function is referred to be continuous at a certain point in its…

Q: If fCxl is continuous on [2,7],then Si f(u) dvz f (7)-(2) True e or false?

A: we have given that f(x) is continuous on[2,7].