1 Let f (x) = ; 1 , then f (x) satisfies the - - 2 conclusion of Rolle's Theorem on 0, 1| 6.

Q: Verify that the three conditions of the hypothesis of Rolle's Theorem hold for the function f (x) =…

A: 1. f(x) is continuous on the closed interval [a,b] 2. f(x) is differentiable on the open interval…

Q: The graph of f(x)= - x2-1 is concave up on interval (-1,1)^O (0,1) 0 (-1,0) U (1, 0 ).co (-∞, – 1)U…

A: Given, fx = xx2-1  We have to find the interval in which graph of  fx = xx2-1 is concave up.

Q: 8. Iff and g are differentiable at xo such that f(xo) g(xo), prove that each of the functions h =…

A: (8) Given that f and g are differentiable at x0 such that fx0≠gx0.Also h=minf,g and H=maxf,g. To…

Q: Suppose that f: [0,1]–R is a function satisfying the conclusions of the intermediate value theorem.…

A:

Q: If f(0) = 0 and f"(x) exists for all x>0, then show that %3D f(x) f'(x) xf"(5), 0 < § < x. Also…

A: Let's find.

Q: 2.if f (x) = x2 + 3, with domain R then f(x) has f(x) %3D O T F

A: Here the given function is f(x)= x2+3 with domain R We need to find it has inverse or not.

Q: 4.1.7. Show that if f : R" + R" satisfies |f(x)| < |r|² for all r € R" then f is differentiable at…

A:

Q: Let f(x) =x sin x and g(x) =x cos x. (a) Show that f'(x) = g(x) + sinx and g'(x) = -f(x) + cos x.…

A:

Q: 1. Examine the continuity and differential possibilities of function f (x)= |x² – 1| in x= 1.

A: Given To check for continuity at x=1 , we find the limit to the left and right of x=1

Q: (a) integrate to find F as a function of x, and (b) demonstrate the Second Fundamental Theorem of…

A: Given- Fx=∫0xtt2+1dt To Integrate-To find F as a function of x, and  demonstrate the Second…

Q: If we apply Rolle's Theorem to the function f(x) = 2x² – 16x – 5 on the interval [2, 6], how many…

A: Given,          fx=2x2-16x-5 on the interval 2,6.We know that,         Rolle's theorem states that…

Q: find the least value of a such that the function f given f(x)=x2+ax+1 is strictly increasing on…

A:

Q: Prove that converting the Higher-Order derivatives to the finite difference formula would be: f(x;)…

A: Solution: (1) f''xi=fxi-2fxi-1+fxi-2∆x2 We know that f'xi=fxi-fxi-1∆x ......(1) Then,…

Q: If a function f(x) = Vx² -1 satisfies conditions of Rolle's theorem on [-1,1] , then the value of…

A:

Q: Define f: [0,2]-R by f(x) = x²+2x-1 , then the sup f- inf f is : [0,2] [0,2] 1.a O V8 b 0 3.cO 8 dO

A:

Q: If A(x) = f(t)dt is increasing on [0,10], then f(x) must be increasing on [O,10]. True False

A: Given

Q: 4. Define f: [0, 1] → R by if x E Qn [0, 1] if x ¢ Qn [0, 1] f(x) = (a) Prove that S° f = 0; s*f =…

A:

Q: The function g(x) = ln(x³ – 1) defined on V2, 0) is not zero unless x 2.

A: Given statement

Q: 1) Consider the function f(x) = In(x + 5) + x + 5 Verify that the function F(x) = x In(x + 5) is an…

A: Explanation of the answer is as follows

Q: If W) is a continuous function on [4,7] and F(x) is an anti-derivative of f(x) such that F(4)=9 and…

A:

Q: 8. Consider the function f(x)=/25-x² . a. Find the linearization L(x) of f(x) at x=3.

A: Given expression is , fx=25-x2 to find out the linearization L(x) of f(x) at x=3.

Q: Show that f (x) = 1 − |x| satisfies the conclusion of the MVT on [a, b] if both a and b are positive…

A:

Q: Let f, g:R R defined by if x 1 (x2 and g(x) = 3x + 2. %3D Find f-1 and g-1.

A:

Q: 1. Find of ахду OF f(x,y) = xy (x+y-1) -x2

A:

Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

A:

Q: 7. Suppose that f: R R is differentiable at c and that f(c) = 0. Show that g(x):= |f(x)| is…

A:

Q: Suppose that a function h satisfies −x^2 ≤ h(x) ≤ x^2 for all x. Prove that h is differentiable at x…

A:

Q: 1. Employ the following methods to find the maximum of f(x) = 4x – 1.8x² + 1.2x – 0.3x* (a)…

A: Given f(x)=4x-1.8x2+1.2x3-0.3x4. To find: (a) find the maximum using golden search method.…

Q: (11) (C) For the function f(x) = x° – x, determine whether the hypothesis of Rolle's theorem hold on…

A:

Q: X if x e Q If f (x) is defined on [0,1] by the rule f (x) = < (1-x , if x 4 Q Prove that fof (x) = x…

A:

Q: 4. Determine if g is differentiable at x = 0 if 2 cos-1 x, if -1<x < 0 g(x) = %3D if x 2 0. x + cos…

A:

Q: Given that F(x) = a) F'(x) bl 5'(4) 2 1 1 + 2 dt.Using the 2nd Fundamental Theorem of Calculus, find

A:

Q: Does the Rolle's Theorem applies for f(z) – ln(2 – z²) over the interval [−1, 1]? O Cannot be…

A: This is a problem related to derivative of the function. Based on the general formula and concept we…

Q: 3. Let f (x) = x ² + 8x + 14. Find all the values interral C-G ,2) such that f (c) = 0 of ċ in the

A:

Q: 1.4 For which values of a eR is the function ar? f(x + iy) = (a'x – axy) +i (- + ay an entire…

A:

Q: b

A:

Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

A:

Q: If f(x) is a continuous function on [4,7] and F(x) is an anti-derivative of f(x) such that F(4)=9…

A:

Q: Find the value of c which satisfies the conclusion of Rolle's Theorem with the given f (x) =x³ - 3x,…

A:

Q: 1. Let f(x) = 4 – 4x?. Verify that f(r) satisfies the conditions of Rolle's Theorem on the interval…

A: 1. The Given function is f(x)=4x-4x2 Diff. with respect to 'x' f'(x)=ln(2).22x+1-8x

Q: For the function f(x)=sin(x)/x defined on [-1,1], there is some -1<c<1 such that f′(c)=0.f′(c)=0.…

A:

Q: Use Part 1 of the Fundamental Theorem of Calculus to find the g'(x) if: g(x) = 1 +t² dt 3x2

A:

Q: 3 - x, x<2 Q.No.15: Is f(x) = {+2 x22 continuous at every x € R. x+1 xs1 -.

A: Continuity of a function.

Q: 1. Consider the function z = f(x, y). If fa(a, b) and fy(a, b) both exist, and they are equal, then…

A: 1. Given: z=fx,y and suppose that fxa,b and fya,b both exists and they are equal and it is…

Q: 15. If f+g is differentiable at a, are f and g necessarily differentiable at a ? If f g and f are…

A: Let f be any function that is not differentiable  at the point a and define as: gx=-fx for all real…

Q: 7. Find all values of x on the graph of f(x)=(x-1)´(2x+1) at which the tangent line is horizontal.…

A:

Q: If ø(x) =(x)+ f(1-x) and f"(x)<0 for all x e increasing in [0,1/2] and decreasing in (1/2,1].…

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.