Given that the function f (x) = ax ^ 2 + 2x is odd, then the real number a is odd

Given that the function f (x) = ax ^ 2 + 2x is odd, then the real number a is odd

In the odd function f (x), f (x) and f (- x) have opposite signs and equal absolute values, that is, f (- x) = - f (x). On the contrary, the function y = f (x) satisfying f (- x) = - f (x) must be odd. For example, f (x) = x ^ (2n-1), n ∈ Z; (f (x) is equal to the 2N-1 power of X, n is an integer)
Therefore, the real number a = 0

The function f (x) = X3 + 2x2 ax is known. For any real number x, f ′ (x) ≥ 2x2 + 2X-4 (I) find the maximum value of real number a; (II) when a is the maximum, the function f (x) = f (x) - x-k has three zeros, and find the value range of real number K

(1) ∵ f ′ (x) = 3x2 + 4x-a, for X ∈ R, f ′ (x) ≥ 2x2 + 2X-4, that is, X2 + 2x-a + 4 ≥ 0, for X ∈ R, it holds ∵ a = 4-4 (4-A) ≤ 0, the solution is: a ≤ 3, ∵ Amax = 3; (2) ∵ a = 3, f (x) = f (x) - x-k has three zeros ∵ k = X3 + 2x2-4x, let g (x) = k, then G

If the equation 2x2 + my2 = 1 represents an ellipse with focus on the y-axis, then the value range of M is______ ．

The ellipse is transformed into the standard equation form, and the equation X212 + Y21M = 1 ∵ represents the ellipse ∵ 1m ∵ 12 with the focus on the y-axis. The solution is 0 ∵ m ∵ 2, and the value range of M is (0,2), so the answer is: (0,2)

As shown in the picture, the circumference of a rectangle is 48 cm. How many cm are its length and width?
The length is five small squares and the width is three small squares

Because length: width = 5:3
So 48 / [(3 + 5) * 2] = 3
So length = 3 * 5 = 15cm
Width = 3 * 3 = 9cm

It is known that sin ((π / 4) - α) = 5 / 13, (0

cos(π/4+α)/cos2α
=√2/2(cosa+sina)/(cos²a-sin²a)
=√2/2(cosa+sina)/[(cosa+sina)(cosa-sina)]
=√2/2*1/(cosa-sina)
sin(（π/4）-α)=5/13,(0

As shown in the figure, AB is the chord of the circle O, and CD is the tangent passing through a point m on the circle O. it is proved that: (1) when AB / / CD, am = MB; (2) when am = MB, AB / / CD

(1) Certification:
When Mo is connected to o, Mn is the diameter
∵ CD is tangent and M is tangent point
∴MN⊥CD
∵AB//CD
∴MN⊥AB
∵ Mn is the diameter
The vertical bisection of Mn ab
‖ am = MB [the distance from the point on the vertical bisector to both ends of the line segment is equal]
(2) Certification:
Take the midpoint n of AB and connect Mn
∵AM=MB
The vertical bisection of Mn ab
The center O is on Mn [the vertical bisector of the string passes through the center]
∵ CD is tangent
∴MN⊥CD
∵MN⊥AB
∴AB//CD

It is known that the function y = (M & # 178; + 2m) x ^ (M & # 178; + m-1) - 2 is a linear function, and its image intersects with the image of inverse scale function y = K / X at a point,
The abscissa of the intersection is 1 / 3, and the analytic expression of the inverse scale function is obtained

Y = (M & # - 178; + 2m) x ^ (M & # - 178; + m-1) - 2 is a linear function, so the degree of X is 1 and the coefficient is not equal to 0, so M & # - 178; + M-1 = 1 gets M = 1 or - 2. When m = - 2, M & # - 178; + 2m = 0, so it should be rounded off, so m = 1, so the abscissa of the intersection of y = 3x-2 is 1 / 3, and y = - 1 is obtained by substituting y = 3x-2, so the intersection (1 / 3, - 1) is substituted into y

Is x = 8 an equation

It's not, it's the solution of the equation

It is known that function f (x) is a positive proportion function, function g (x) is an inverse proportion function, and f (1) = 1, G (1) = 2. Find the range of F (x) + G (x)

Because the function f (x) is a positive proportional function, Let f (x) = KX (k is not equal to 0)
Because the function g (x) is an inverse proportional function, let g (x) = t / X (t is not equal to 0)
If (1) = 1, G (1) = 2, then k = 1, t = 2
Then f (x) + G (x) = x + 2 / X
The denominator is obviously not = 0, so the range is (- infinity, 0) U (0, + infinity)

How much is 3 / 25 times 101 minus 3 / 25

=3 / 25 × (101-1)
=3 / 25 × 100
=12