What is the way to prove that a curve is a circle?

What is the way to prove that a curve is a circle?

It is proved that the distance from any point of the curve to a fixed point is equal

It is proved that if all the close planes of a curve pass through a fixed point, it is a plane curve

Let t (s) = R '(s) be tangent vector (velocity vector) and n (s) = R' '(s) / | R' '(s) | be unit vector in principal normal direction

How to prove that a curve is symmetrical about a point

The meaning of the symmetry of two points a and B about the origin is that the sum of the corresponding coordinates of a and B is equal to zero

What is LM Curve?
LM Curve is derived from the relationship between speculative demand and interest rate of money, the relationship between transaction demand and prudent demand (i.e. preventive demand) and income, and the relationship between money demand and supply. The graph of the relationship between income y and interest rate r satisfying the equilibrium condition of money market is called LM Curve, Any point on the LM Curve represents a certain combination of interest rate and income. Under such a combination, money demand and supply are equal, that is, money market is in equilibrium

LM Curve is used to describe the relationship between national income and interest rate in the equilibrium state of money market
LM Curve represents the locus of the points of various combinations of income and interest rate when money supply equals money demand in money market. The mathematical expression of LM Curve is m / P = ky-hr, and its slope is positive, which indicates that LM Curve is generally a curve inclined to the upper right, All of them are the unbalanced combination of money demand greater than money supply; the combination of income and interest rate on the left side of LM Curve is the unbalanced combination of money demand less than money supply; only the combination of income and interest rate on LM Curve is the balanced combination of money demand equal to money supply
LM Curve is a curve described by different combinations of income and equilibrium interest rate, which makes the money market in equilibrium. In other words, every point on LM Curve represents the combination of income and interest rate, which just makes the money market in equilibrium

Find the inflection point coordinates of the curve y = (x-1) / (x ^ 2 + 1)
Two derivations y '' = 0 i know

y ' = 1/(x^2+1) + (x-1) * (-2x) / (x^2+1)^2= (1+2x-x^2) / (x^2+1)^2y '' = (2-2x) / (x^2+1)^2 + (1+2x-x^2) * ( - 4x) / (x^2+1)^3= 2 ( x^3 - 3x^2 - 3x +1 ) / (x^2+1)^3y '' = 0 => x^3 - 3x^2 - 3x +1 = 0...

Find the concave convex interval and inflection point of the curve y = (x-1) multiplied by & # 179; √ X & # 178

Derivation first
y‘=³√X² +（2/3）×（x-1）×x^(-1/3)
Then, the derivative y '' = (2 / 3) x ^ (- 1 / 3) + (2 / 3) x ^ (- 1 / 3) - 2 / 9 (x-1) x ^ (- 4 / 3) > 0, and the solution x ＞ - 0.2
When y '' = (2 / 3) x ^ (- 1 / 3) + (2 / 3) x ^ (- 1 / 3) - 2 / 9 (x-1) x ^ (- 4 / 3) < 0, the solution is x ＜ - 0.2
So the inflection point of the curve is obtained when x = 0.2, the concave interval of the curve is (- 0.2, + ∞) and the convex interval is (- ∞, - 0.2)

(-3x²y³）（x²-1）-（x²-1）×5x²y³

(-3x²y³）（x²-1）-（x²-1）×5x²y³=(x²-1)(-3x²y³-5x²y³)
=-8x²y³(x+1)(x-1)

Determine the concave convex interval and inflection point of F (x) = 3x ^ 5 / 3 + 5 / 3x ^ 2

f'(x)=5x^(2/3)+(10/3)x
f''(x)=(10/3)x^(-1/3)+10/3=0
x^(-1/3)=-1
x=-1
X ^ (- 1 / 3) is a decreasing function, and X ≠ 0
So x0
-1

Finding tangent equation of curve
How to find the tangent equation of curve (5Y + 2) ^ 3 = (2x + 1) ^ 5 at point (0, - 1 / 5)

Take 1 / 3 power on both sides, the left side is 5Y + 2, then subtract 2 from both sides, divide by 5, and the left side is y. take the derivative on the right side and calculate the derivative expression. When x = 0, the derivative value is 2 / 3 (that is, the slope), and the straight line passing through (0, - 1 / 5) is y + 1 / 5 = 2 / 3 X

If a point on the curve is known, the tangent equation of the point can be obtained
There are two cases, one is that the point is tangent point, the other is that the point is not tangent point
When the point is not a tangent point, another tangent point will be obtained, but there can only be one tangent point between two curves. If this point has been passed, where is the other tangent point

The definition of tangency is not that a line and a curve have a common point,
The definition of tangent is the limit position of secant,
For example, if y = SiNx, then y = 1 is the tangent of a curve, but there are countless intersections