Let the function z = XY + e ^ XY, find the directional derivative of 1 function along the negative X axis at (1,1); 2 the directional derivative of the function at (1,1) along the vector (1,2); Maximum directional derivative of 3 function at (1,1)

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Let the function f (x) = INX + x ^ 2-2ax + A ^ 2, a belongs to R, and find the extreme point of F (x) The teacher suggested that we can discuss the relationship between X = A / 2 and Y axis, but I forgot how to do it

Domain x > 0
f'(x)=1/x+2x-2a=1/x*(2x^2-2ax+1)=1/x*[2(x-a/2)^2+1-a^2/2]
When 1-A ^ 2 / 2 > = 0, that is - √ 2 = √ 2, X1 > 0 and X2 > 0 are extreme points
When a

Let f (x) = lnx-1 / 2aX ^ 2-bx 1. When a = b = 1 / 2, find the maximum value of F (x) 2. On the other hand, f (x) = f (x) + 1 / 2aX ^ 2 + BX + A / X (0 < x ≤ 3) takes the slope k ≤ 1 / 2 of any point P (x0, Y0) on the image as the tangent point, and finds the value range of real number a 3. When a = 0 and B = - 1, equation 2mf (x) = x ^ 2 has a unique real number solution, and find the value of positive number M

1.f(x)=lnx-1/4 x^2-1/2 x ,x>0
The derivative is f '(x) = 1 / X-1 / 2 X-1 / 2; F '' (x) = - 1 / x ^ 2-1 / 2 = - x ^ 2 / 2, G (x) is 0 at the maximum value of (0,3], so a > = 3
3.f(x)=lnx+x,x>0.
2m (LNX + x) = x ^ 2 has a unique solution

On the problem of function. It is known that the function f (x) = 2aX + B / x + INX. (1) if the function f (x) obtains the extreme value at x = 1 and x = 1 / 2, find the values of a and B (2) If f '(1) = 2, the function f (x) is a monotone function on (0, positive infinity), find the value range of A

(1) F '(x) is: 2a-b / x ^ 2 + 1 / x = (2aX ^ 2 + X-B) / x ^ 2
F (x) gets the extreme value at x = 1 and x = 1 / 2, so f (x) '= f' (1 / 2) = 0
2a+1-b=0,a/2+1/2-b=0
a=-1/3,b=1/3
(2) F '(1) = 2, then 2A + 1-B = 2,
It is monotonically increasing in the definition field,
So f '(x) > = 0, then a > 0 and 1 + 8ab < = 0
1+8a(2a-1)<=0,（4a-1)^2<=0
a=1/4

Find the monotone interval and its extreme value of function f (x) = x / INX

The increasing interval is (E, positive infinity), the decreasing interval is (0, e), and the minimum is e

F (x) = INX ax, find the extreme point of function f (x) Second, when a is greater than 0, there is always f (x) less than or equal to - 1. Find the value range of A

F '(x) = 1 / x-a = 0 gets x = 1 / a because f' '(x) = - 1 / X ²

Let the function z = XY + e ^ XY, find the directional derivative of 1 function along the negative X axis at (1,1); 2 the directional derivative of the function at (1,1) along the vector (1,2); Maximum directional derivative of 3 function at (1,1)