Suppose f: [a,b] → IR is continuous on [a, b] and differentiable on (a,b) such that f'(x) 2 0 fur all xt (a,b). Show that if 3 x € (a, b) st. f'xo) > 0 , then flbl > fla).

Q: Suppose f:R→IR is a diffexetiuble function which satısfies flo) = 0 . We wort to shouw f': IR → IR…

A: We have to do implicit differentiation.

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)→R be de fined by X€QN(a,b) x€Q' n(a,b) -2 f(x)=' 3

A: This is a problem of Differential Calculus.

Q: Use Green's Thoerem to evaluate F dr. where F(r, y) = (-3Vx² + 2, 7 tan '(x)) and C is the triangle…

A: We have to evaluate by Green's theorem.

Q: KB is ∀xy F(x, y). Prove using resolution-refutation that ∀xy F(y, x).

A: Solution: First, we convert clauses to CNF : (1) ∀xy Fx,y from KB (2) ∀x Fx,Ax (3) Fx,Ax CNF

Q: Show that A Function with a Single Point of Discontinuity 2xy x² +1,²² 0, is continuous at every…

A:

Q: 1. Evaluate: 1 1+x 2. Find Fa) given f'a)=3Sinx-4e*, FET)=4e"

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Q: f 'C) =7x*+7-3 ond fl0) 3D8

A:

Q: suppose that f (u, v) = ue", where u =rs and v =r + s. The composite function f(u, v) is equal to:…

A: A composite function is one which is made of other functions or in other words function values are…

Q: if f is derivable in [a, b] and f' (a) not equal to f' (b), then for each number k lying between f'…

A:

Q: Assume that f : (0, 1) → R is a differentiable function with the derivative f' increasing on (0,…

A: Given f:0,1→R is a differentiable function and derivative of f is increasing in 0,1 that is f'x>0…

Q: Is there a twice differentiable function f R2R whose best adratic approximation at the point (1,0)…

A: Given problem is from differential calculus in several variables. We have shown that there exists a…

Q: Assume that f(x, y) is a function given by if (r, y) # (0,0), f(x, y) 0 if (r, y) = (0, 0). Page 1…

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Q: Suppose f, g:C → C are differentiable. Prove that fg is differentiable and that (fg)' = f'g + fg'.

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Q: Determine whether the statement is true or false. '8 If f'is continuous on [7, 8], then f '(v) dv =…

A:

Q: Let f : R2 → R with f(x,y) = px^2 + y^2. Sketch the graph of f. Prove that f is not differentiable at…

A: Given f(x,y)=px2+y2

Q: Find region E into the spherical region B Select one: 0 а. бл O b. 2 O c. 4 O d. 8r e. + +이 dV,…

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Q: Assume ƒ is a continuous on [2, 4],, and ƒ(2)= 4. Evaluate ƒ(4).

A: we have been given that f is continuous on [2,4] and f(2)=4. we have to evaluate f(4). the given…

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: Let & be continuous in [a,b] and differentiable in (a,b). It's true that: a) f (a) = f(b) = 3 x €…

A: Statement (a): f(a)=f(b)  implies there exists x0​∈(a,b)  such that the tangent line to the graph of…

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: d) Let f be a continuous function. Then 1 [ S f(x,y) dady = f*f* f(x,y) dyda. 0

A:

Q: Let f be a function with continuous second derivatives, defined by: z = f (xy, x² + y") If we take u…

A: Solution:-

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: 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].

Q: Suppose that f: R² R is a function given by y³ x² + y² f(x, y) = 0 for x = y = 0. Show that f(x, y)…

A:

Q: If f' is continuous on [-1,4], then S*, f'(x)dx = f(4) – f(-1) Select one: O True O False

A: Answer Is TRUE

Q: 3. Let there be some a, b R such that f(a)ƒ'(a) = f(b)ƒ'(b) + 1. Show that f f(x) f"(x)dx ≤ 1.

A: This is a problem related to integration. Based on the general formula and concept we will prove the…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.