two positive numbers x and y, with the maximum value 4, add up to 5. what is the difference between the maximum and minimum valu
e of the product x2y3
1 answer:
Answer:
92
Step-by-step explanation:
Since the sum of the two numbers is 5, we can represent one of them by x and the other by 5-x. Then the desired product is ...
x²(5-x)³
A graphing calculator can show the extreme values of this on the interval 1 ≤ x ≤ 4. The maximum is 108 at x=2; the minimum is 16 at x=4.
The difference between the maximum and minimum is 108-16 = 92.
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If you like, you can take the derivative and set it to zero.
f(x) = x²(5 -x)³
f'(x) = 2x(5 -x)³ +x²(-3)(5-x)² = x(5 -x)²(2(5-x) -3x)
f'(x) = 5x(5-x)²(2-x)
This will be zero for x=0, x=5, and x=2. The points at x=0 and x=5 represent minima in the product. The values x=0 and x=5 are not in the domain of interest. The point at x=2 represents a maximum.
To find the function extremes on an interval, we need to evaluate the function where the derivative is zero, and also at the ends of the interval. So, the function values of interest are ...
f(1) = 1²·4³ = 64
f(2) = 2²·3³ = 108 . . . . product maximum
f(4) = 4²·1³ = 16 . . . . . . product minimum
The difference between the maximum and minimum is 92.
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Answer:
To this question; we want to show that if two antipodal points on a sphere were A and B, for any random point C on the sphere, AC is perpendicular to BC?
Step-by-step explanation:
I would proceed by imagining the sphere in a three dimensional Cartesian coordinate system. For example, use a sphere of diameter 2 and let it sit at the origin. Then (0, 0, 1) and (0, 0, -1) are the vector locations of the “north and south poles” of the sphere.
Now choose the vector location of any point on the surface of the sphere. It will have a vector location - we’ll call it (x, y, z). Now the vectors from your point to the two poles are (-x, -y, 1-z) and (-x, -y, -1-z).
Now just form the dot product of those two vectors:
(-x, -y, 1-z) . (-x, -y, -1-z) = x^2 +y^2 + (1-z)*(-1-z)
Now the truth of your claim will be embodied in that dot product being zero:
x^2 + y^2 - (1+z)(1-z) = 0
x^2 + y^2 - (1-z^2) = 0
x^2 + y^2 + z^2 = 1
But that last line is just the definition of points on the surface of a sphere of radius 1, so the claim is proven.
Since R>0 and AC is perpendicular to BC, the <ACB is at right angle.
Please, find attached a simple image to show antipodal points (Two points that makes a diameter) and a point C on the sphere.
