What is the sufficient necessary condition for the function f x to be derivable at x 1?

The sufficient and necessary conditions for derivability are the existence of a limit of the tangible [f(x+a)-f(x-b)] (a+b), which satisfies a, b and is independent of each other, and can be positive or negative.

A obviously can't, because h is equal and does not satisfy the irrelevant conditions of A and B.

b obviously not, because ln(1+2h2)>0, cannot go to the 1- direction.

c is also not good, because the 1-cosh part of 1+1-cosh is always greater than 0 only d is satisfied, and.

e h 1+h, the limit is equal to f'(1)

Let t=2h, then h=t 2, when h 0, t 0 then lim(h 0)[f(a)-f(a+2h)] hlim(h 0)2[f(a)-f(a+2h)] 2hlim(t 0)2[f(a)-f(a+t)] t-2lim(t 0)[f(a+t)-f(a)] t-2f'（a）

So if the limit of lim(h 0)[f(a)-f(a+2h)] h exists, then f'(a) Existence.

And f'(a) exists, then the limit of lim(h 0)[f(a)-f(a+2h)] h exists.

So this is indeed a sufficient and necessary condition for f(x) to be derivable at x=a.

Question 4 is correct.

The reason why a is wrong is that it does not conform to the definition of derivative, and it can be seen that the other three equations have the term -f(1), and f(1) does not appear in a, so it must not be related to f(x) at x=1.

The problem with b is that H2, having a square means that the amount of change is positive, and ln(1+H2) is equivalent to H2, which only exists in the right derivative of x=1, and the left end cannot be guaranteed.

The problem of c is similar to that of b, where 2-cosh—1=1-cosh, and 1-cosh is equivalent to h2 2, and also appears squared, with only one derivative present.

Therefore d is correct and e h-1 is equivalent to h

If f(a)>0, then in the neighborhood of x=a, there is |f(x)|=f(x), the derivative of which is f'(a)

If f(a)>0, then in the neighborhood of x=a, there is |f(x)|=f(x), the derivative of which is f'(-a) If f(a)=0, if in the neighborhood of x=a, f(x) does not change the sign, then f(a) is the extreme point, and there is f'(a)=0, then |f'(a)|=0

If f(a)=0, but in the neighborhood of x=a, f(x) is described, then f(a) is not an extreme point, f'(a) ≠0, at this time|f'(a)|The left and right derivatives of are f'(a) and the other is -f'(a), the two are not equal, so x=a is not derivable.

In summary, |f(x)|The sufficient condition for the non-derivative at x=a is: f(a)=0, but f'(a)≠0.

Summary. Hello, by f(x) at the point x0 derivable, get.

f′(x0)=

limx→0yx

Hence y x

f (x0) + where.

lim α=0

x→0△y=f′（x0）△x+α△x

And x=o(x), and f(x0) does not depend on x, so by the definition of differentiation, we know that f(x) is differentiable at the point x0 If f(x) is differentiable at the point x0, then.

y=a△x+o（△x）

y/△xa+o(△x)/△x

a=lim △y/△x

x→0f(x0)

i.e. f(x) is derivable at the point x0.

Therefore, the function f(x) is derivable at the point x0 and is a sufficient and necessary condition that is differentiable at that point, and the function f(x) is derivable at the point x0 is f (sensitivity x) and the differentiable condition () at the point x0 is a sufficient condition b necessary condition c sufficient necessary condition difference closed d

You have already counted well, and f(x) is derivable at point x0, so that f (x0)=lim x 0 y x Therefore, y x=f (x0)+ where lim =0 x 0 y=f (x0) x+ x and x=o(x), and f (x0) does not depend on x, so by the definition of differentiation, you can know that f(x) is differentiable at point x0 If f(x) is differentiable at point x0, then y=a x+o( x) y x=a+o( x) x a=lim y x x 0=f(x0) i.e. f(x) At the point x0 the derivable function f (scatter x) is derivable at the point x0 is a sufficient and necessary condition for differentiability at that point

Okay, thanks. You're welcome

Let f(x) be somewhere in x=aNeighborhood, then f(x) is a derivable one at x=aSufficient conditionsYes (d).

Sufficient and necessary conditions for the function to be derivable.

The function is continuous at that point and both the left and right derivatives exist and are equal.

Description: The derivative function is continuous; Function continuity is not necessarily derivable; Discontinuous functions must not be derivative.

Derivative Properties:

Not all functions have derivatives, and a function does not necessarily have derivatives at all points. If a function exists at a certain handicap derivative, it is said to be derivable at that point, otherwise it is called non-derivative. However, the derivable function must be continuous; Discontinuous functions must not be derivative.

For the derivative function f(x), x f'(x) is also a function called a derivative of f(x). The process of finding the derivative of a known function at a point or its derivative is called derivative. In essence, derivative is a process of finding the limit, and the four rules of the derivative are also the four rules of the limit.

Conversely, a known derivative can also be reversed to find the original function, i.e., an indefinite integral.

f(x) is the one that is derivable at x=aSufficient conditionsYes

This title defines the basis as long as lim[f(a)-f(a-h)] h exists (h tends to 0).

A domain of x=a is [a-h,a+h], and the h region is zero.

The derivative is also called the derivative function.

Value. Also known as micro-business, it is calculus.

important foundational concepts in . When the independent variable x of the function y=f(x) produces an increment δx at a point x0, the ratio of the increment δy of the output value of the function to the increment δx of the independent variable is at the limit a when δx approaches 0 If there is a hailstorm, a is the derivative at x0 and is denoted as f'(x0) or df(x0) dx.

Origin:

Around 1629, the mathematician Fermat of the French Envy State.

The tangent of the curve is studied.

and methods for finding the extreme value of a function; Around 1637, he wrote a manuscript entitled "Methods for Finding Maximums and Minimums".

In making the tangents, he constructed the difference.

f(a+e)-f(a), the factor e found is what we call the derivative f'（a）。

If the function y=f(x) is derivable at the point x0, then the function f(x) must be continuous at the point x0; If the function y=f(x) is continuous at the point x0, then f(x) may not be derivable at the point x0; But if y=f(x) is discontinuous at point x0, then y=f(x) must be underivable at point x0. Therefore, y=f(x) is a derivable sufficient and necessary condition at the point x0.

is y=f(x) continuous at point x0. The derivative is defined as: [f(x)-f(x0)] x-x0) at the limit of x approaching x0, if it exists, denoted as the derivative of f at x=x0.

In this extreme brightness limit, x tends to x0 and can be approached from the right or left side of the key. The corresponding limits are the right and left derivatives, respectively.

According to the limit. Definition, the upper and outer limits exist and are equivalent to the left and right limits both exist and are equal!

Closed interval. The endpoint of the terminal has only one derivative, for example, the left endpoint has only the right derivative. Sometimes I don't emphasize this difference for fear of trouble.

Otherwise, every time it is said, "f(x) is derivable on (a,b), has a right derivative at point a, and has a left derivative at point b." ”

Summary. Affinity If the function f(x) is derivable at the point x0, then |f（x）|Select c. at point x0Continuous but not derivable.

If a function is derivable at a certain point, then it must first satisfy the condition of continuity at that point, so that the derivability must be continuous. But continuity is not necessarily derivable, such as some piecewise functions.

If the function f(x) is derivable at the point x0, then |f（x）|At point x0? a.Leadible bUnconductable cContinuous, but not necessarily derivable.

Affinity If the function f(x) is derivable at the point x0, then |f（x）|Select c. at point x0Continuous, but not spine-based. If a function is derivable at a certain point, then it must first satisfy the condition of continuity at that point, so the derivability must be continuous.

But continuity is not necessarily derivable, such as the first piecewise functions.

So if the function f(x) is derivable at the point x0, then the deduce is |f（x）|is continuous at point x0 but cannot be derivable. (The same is true for absolute values).

I sincerely hope mine is helpful to you

For a function to be derivable, the left and right derivatives must first be equal.

Second, there should be a definition at that point.

A sufficient condition for f(x) to be derivable at x=a is the existence of lim(x approaching 0) [f(a)-f(a-h)]h.

1) If the derivative is greater than zero, it increases monotonically; If the derivative is less than zero, it decreases monotonically; A derivative equal to zero is a stationary point of the function, not necessarily an extreme point. It is necessary to substitute the values on the left and right sides of the settlement point to find the positive and negative derivatives to judge the monotonicity.

2) If the function is known to be an increasing function, the derivative is greater than or equal to zero; If the function is known to be decreasing, the derivative is less than or equal to zero.

For a univariate function, continuity is a necessary condition for differentiability, that is, differentiability must be continuous. The differentiability is a sufficient condition of continuity, that is, the differentiability must be continuous.

For a univariate function, derivability and differentiability are equivalent, that is, derivable and differentiable are sufficient and necessary conditions for each other.

But this is not true for multivariate functions, the existence of partial derivatives is not necessarily differentiable, and the partial derivatives of differentiable multivariate functions must exist.

b, the unary function is derivable and differentiable.

Continuity is differentiableNecessaryThe condition is not sufficient.

Conductivity is a sufficient and necessary condition for differentiability to (y'=dy/dx)

DifferentiableYesContinuous sufficient conditions are notNecessaryCondition.