Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is Lipschitz continuous on (−∞, ∞).

Q: There exists a differentiable function f (-1, 1)→ R that does not have any local extrema but whose…

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Q: Let f: R → R be differentiable. Suppose that the first derivative f' atisfies |f'(y) – f'(x)| < L\y…

A: Given, f'y-f'x≤Ly-x. To show fy-fx-f'xy-x≤Ly-x2.   Consider , fy-fx-f'xy-x=fy-fxy-x-f'x=f'x-f'x=0.…

Q: Determine whether the function xy²/³ + xy ³√x⁹ + 2y³¹ 1 √3' if (x, y) = (0,0) if (x, y) = (0,0) is…

A: Let us consider a function f then f is said to be continuous at c if f(c) is defined limx→cf(x)…

Q: Show that the limit of the function f(x,y)= (0,0). (x+y)² x² + y² is indeterminate at

A: Limit of the function

Q: If we have that f: R → Ris a continuous function , in which f'(7) = 6 and f(7) = -2 an equation of…

A: We have to find out tangent line

Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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Q: Let f be a function of R sucht hat Ә f(x, y) = f(x,y); for all (x, y) = R then prove that - y^ f(x,…

A: Since given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation for…

Q: Let ƒ : [a, b] → R be differentiable on [a, b]. Let λ = R be such that ƒ'(a) < \ < f'(b). (i) Let h…

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Q: Assume that f(r, y) is a function given by f(x, y) = x² + y² + x?y+4 %3D on a bounded domain D =…

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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 <…

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Q: Let z=x²y, where x(s,t) and y(s,t) are differentiable functions with values Әx Əs x(2,0) = 4, y(2,0)…

A: Given : z = x^2.y

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

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Q: 3. Let z = yg(x2 - y²) be a differentiable function. Show that z(x,y) satisfies 1 дz 10z 7 - + - = х…

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Q: Use second-order partial derivative characterization to determine if The following functions f: Rn…

A: a) f(x,y)=x+3xy+6x2+y2 is strictly convex.b) f(x,y)=−x2−2xy−y2 is neither strictly convex nor…

Q: Let f(x, y) and let g(x, y) = O O 8xy8 828+8y8 O 3xy6 3x² + 3y¹2 Which of the above functions are…

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Q: Let F(x , y , z) = g(y) + h(z) + 5 In(x - 2) , where g(y) and h(z) are differentiable functions of…

A: If a function is a function of more than one independent variable then the derivative of this…

Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

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Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

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Q: Calculate the divergence of cach of the following functions (a) r²î + y²§ + 2²k. (b) yzî + xzj+ xyk.…

A: As per the instruction, we will answer only the first 3 subparts. For (a), For a given vector fx, y,…

Q: 6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) =…

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Q: If F(t) = 2xyi + xyz2j + ex+yk, then what is the divergence of F? Select one: A. 2yi + xz2j + ex+yk…

A: The divergence of a vector F=P,Q,R is given by div F=∂P∂x+∂Q∂y+∂R∂z. The given vector is Ft=2xy…

Q: Let a differentiable function z = f(h²(x² + y3), 2łn(x +y)) satisfying the condition f(4,0) 1 be…

A: We need to find the partial derivative of z with respect to x for the given function.

Q: (b) 22). Compute the partial derivative frryz for f(x, y, z) = cos(cos(x³ + z²)) + x² In(y² +

A: Given f(x,y,z)=cos(cos(x3+z2))+x2ln(y2+z2).We need to find the partial derivative fxxyz for the…

Q: Let I be and interval and let f : I →R and g : I → R be functions differentiable at c. If h : I → R…

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Q: A bounded, 3B > 0 such that |f(x)| < B Vx € (0, 1), continuous function on (0, 1) with unded…

A: We will find an example

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: