If f is increasing only if f'(x) ≥ for all x ∈ I, and if f is decreasing only if f'(x)≤0 for all x ∈ I   If f is increasing, then for all x and c in I we have ((f(x)-f(c)/(x-c))≥0. Taking a limit as x goes to c, we see that f'(c)≥0.   Then suppose f'(x)≥0 for all x ∈ I. Let x < y in I.  Then there is some c ∈ (x, y) such that f(x)-f(y)=f'(c)(x-y).   f'(c)≥0.  x-y<0, so f(x)-f(y)≤0   so f is increasing.   Show f is decreasing if f'(x)≤0 for all x ∈ I

Q: x if x20 If Ax) = -x2 then f'(0) if x <0

A: Given function, fx=x2      if x≥0-x2   if x&lt;0

Q: Suppose that f(x) Evaluate each of the following: f'(3) = %3D f'(-1) =

A: Given function is : fx=3x9 The given function can be written as : fx=3x-9

Q: (a) In C, we have f "(x) &lt; 0 for all x . (b) In A, f "(x) goes from+ to-. (c) In B, we have f…

A: These statements refer to the second derivative of a function f(x) and describe how the function is…

Q: Suppose that ƒ′(x)&lt;= 1 for 1&lt;= x&lt;= 4. Show that ƒ(4) - ƒ(1)&lt;=3.

A: Show that:

Q: Q8. We shall formally show that if two functions f and q are both differentiable at c, then we have…

A:

Q: Let f be defined by the graph to the right. Find VK0.3)+ 1(1.1) - x) +1(-1)+1(2)-1(-*)

A:

Q: Suppose that f (0) = 2 and f'(x) 0. Show that f(4) 0.

A: f(0)=2   and f'(x)≤3  for all x&gt;0 ⋯(i)integrate (i) w.r.to x ,we getf(x)≤3x+kput…

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A:

Q: f(x) = x(3x-(9x2+2)1/2). Compute the limit of f(x) as x tends to positive infinity.

A: Given data:   The expression for the given function is f(x)=x(3x-(9x2+2)1/2).   The expression for…

Q: Find the partial derivative for the function a. b. Əz C. əz O2 = -5y² sin(x² + y²) e. d. əz dy +-5y6…

A:

Q: Is the following set of vectors S = {v,, V2. V3} basis for R³ space? Where, a. v = (3,1,–4), v2 =…

A:

Q: X = (1-_-_-_-)* 2 o then f(x) tend:

A:

Q: 6. The function f(x) is created by doing a vertical compression of 3 on g(x). If (7, 3) is a…

A: Solution-

Q: 6. The function f(x) is created by doing a horizontal compression of 3 on g(x). If (7, 3) is a…

A: horizontal compression means horizontal stretch.

Q: Suppose f is a function with f(2)=5 where for all interval > 0, there is some δ > 0 such that if x…

A: Answer

Q: 1). Does the left-side limit of the difference quotient of the function g(w)=|w| exist at w=0? If it…

A:

Q: Let f (x) = ln (arccot (x*)). What is f' (x)? 4x3 (1+ x8 ) arccot(x4) 3x3 (1+x7) arccot(x4) 4x3…

A:

Q: Let g: RxR → RxR be a function defined by g(x, y) = (x², x-y). Suppose that X = Rx{0} and D= {(a, a)…

A: Bartleby Guideline : ''Since you have posted a question with multiple sub parts , we will provide…

Q: Let  f (x) be a function which is continuous on [−7, 3] and differentiable on (−7, 3). Suppose that…

A:

Q: a) State the 4 ayions of an inner product space. Cauchy Schonty inequality b) State the Cauchy-…

A: Here,  we have to state axioms of inner product space and cauchy Schwartz inequality.

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A:

Q: If we need to calculate the derivative of a function at a point, there are two ways we can think…

A: Given: gx=x-x2

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.