Find the minimum value of the function f (x) = root sign (x ^ 2 + y ^ 2) + root sign ((x-1) ^ 2 + y ^ 2) + root sign (x ^ 2 + (Y-1) ^ 2) + root sign ((x-3) ^ 2 + (y-4) ^ 2)

F (x) denotes the sum of distances from point P (x, y) to four points a (0,0), B (1,0), C (0,1) and D (3,4)
The shortest distance is the sum of line AD and line BC
Therefore, the minimum value is root (3 ^ 2 + 4 ^ 2) + root (1 ^ 2 + 1 ^ 2) = 5 + root 2

Explore the maximum and minimum value of 5-x under the root sign X-1 + 3 times the root sign of the function y = 4 times Why do you use 4 times root sign X-1 = 3 times root sign 5-x to determine the maximum value

Let f (x) = 2 / √ (x-1) - 3 (x-1) - 3 (5-x) x ∈ [1,5] 2. Let f '(x) = 2 / √ (x-1) - 3 / (2 √ (5-x)) = 03 √ (x-1) = 4 √ (5-x) = > 9x-9 = 80-16x = = > x = 89 / 25F (1) = 4 √ (1-1) + 3 √ (5-5-1) = 6F (89 / 25) = 4 √ (89 / 25-1) + 3 √ (5-89 / 25-1) + 3 √ (5-89 / 25-1) + 3 √ (5-89 / 25-1 / 5-89 / 89 / 25-1) + 3 (5-89 / 89 / 89 / 25-1), 3 √25) = 10F (5) = 4 √ (5-1) + 3

How to find the maximum and minimum of the known function y = root (1-x) + root (x + 3)

Obviously Y > = 0
So both sides square
y^2=1-x+2√(1-x)(x+3)+x+3
=4+2√(-x^2-2x+3)
=4+2√[-(x+1)^2+4]
Domain defined by
1-x>=0,x+3>=0
So - 3

If the function f (x) = ax (a > 0, and a ≠ 1) has a maximum value of 4 and a minimum value m on [- 1, 2], and the function g (x) = (1-4m) If x is an increasing function on [0, + ∞), then a = () A. 1 Two B. -1 Two C. 1 Four D. 4

Let g (x) = (1-4m) x be an increasing function on [0, ∞], then 1-4m > 0 is obtained, and m < 14 is obtained. ① if a > 1, then f (x) increases on [- 1, 2],  f (x) max = f (2) = A2 = 4, the solution a = 2, f (x) min = 2-1 = 12 = m, which is inconsistent with m < 14; ② 0 < a < 1, then f (x) decreases on [- 1, 2]  f

Given that the maximum value of the function y = 1-x + X under the root sign x + 3 is m and the minimum value is m, then M / M =? Yes (radical 2) / 2

1-x>=0 x=0 x>=-3
-3

Find the maximum and minimum value of the function y = 1-x + X + 3

Firstly, we know that the value range of X is: - 3 ≤ x ≤ 1
Secondly, square the two sides of the original formula to obtain:
y^2＝1－x＋2√[(1－x)(x＋3)]＋x＋3
＝4＋2√(3－2x－x^2)
＝4＋2√[4－(x＋1)^2]
It can be seen from - 3 ≤ x ≤ 1: - 2 ≤ x + 1 ≤ 2
∴0≤(x＋1)^2≤4
∴0≤4－(x＋1)^2≤4
∴0≤2√[4－(x＋1)^2]≤4
∴4≤y^2≤8
Obviously Y > 0
∴2≤y≤2√2

Find the minimum value of x ^ 2 + 3 under the function y = x ^ 2 + 4 / root sign

Let a = √ (x 2 + 3)
Then a ≥ √ 3
And x 2 + 4 = a 2 + 1
So y = (a 2 + 1) / a = a + 1 / A
This is a tick function, when a > 1 increases
Here a ≥ √ 3
So the minimum value = √ 3 + 1 / √ 3 = 4 √ 3 / 3

Let f (x) = asin (x + π / 3) - ((radical 3) / 2) cosx, and f (π / 3) = (radical 3) / 4. (1) find the value of real number a; (2) find the minimum positive period and monotone increasing interval of function y = f (x) × cosx

∵f(π/3)=√3/4,
∴f(π/3)=asin[(π/3)+(π/3)]-(√3/2)cos(π/3)
=a-(√3/4）
∴√3/4=a-(√3/4)
The solution is: a = √ 3 / 2

The known function f (x) = radical 2 asin (x - π / 4) + A + B When a < 0, the value range of F (x) on [0, π] is [2,3], and the values of a and B are obtained

0≤x≤π
-π/4≤x-π/4≤3π/4
sin(x-π/4)∈【-√2/2,1】
ab=3
The minimum value √ 2 A * 1 + A + B = 2 -------- a = - 1 / (√ 2 + 1) = 1 - √ 2
a=1-√2,b=3

Given the function f (x) = radical 3 * SiNx / 4 * cosx / 4 + cos ^ 2 * x / 4 + 1 / 2 (1), the analytic formula, period and symmetry center of F (x) are obtained

F (x) = radical 3 * SiNx / 4 * cosx / 4 + cos ^ 2 * x / 4 + 1 / 2 = ((radical 3) / 2) SiNx / 2 + (cosx / 2 + 1) / 2 + 1 / 2 = ((radical 3) / 2) SiNx / 2 + (1 / 2) cosx / 2 + 1 = cos30 ° SiNx / 2 + sin30 ° cosx / 2 + 1 = sin (x / 2 + 30 °) + 1 the period is 4 π, and the symmetry center is (- π / 3,1)

Find the minimum value of the function f (x) = root sign (x ^ 2 + y ^ 2) + root sign ((x-1) ^ 2 + y ^ 2) + root sign (x ^ 2 + (Y-1) ^ 2) + root sign ((x-3) ^ 2 + (y-4) ^ 2)