Hello! Really can't wrap my head around this so please help!
Consider the function:
f(x) = x^2+(1000-x)^2
First find out where it is increasing, and then use this fact to determine which number is larger:
a) 1000^2
or
b) 998^2 + 2^2
I found that the function increases at x=500 but how do I use that fact to figure this out???
f(x) = x^2+(1000-x)^2
= 2x^2 - 2000x + 1000000
f' = 4x - 2000
f'=0 at x=500
f' < 0 for x < 500, so f is decreasing
f' > 0 for x > 500, so f is increasing
so, f(1000) > f(998) because f is increasing at that point
f(x) = x^2+(1000-x)^2
... = x² + 1E6 - 2000x + x²
... = 2x² - 2000x + 1E6
the minimum is on the axis of symmetry at x=500 ... the function increases from there
so as x increases (below x=500), the function decreases
f(0) = 0² + (1000 - 0)² = 1000²
f(2) = 2² + (1000 - 2)² = 998² + 2²
To determine where the function f(x) = x^2 + (1000 - x)^2 is increasing, you need to find its derivative and identify the values of x where the derivative is positive.
Let's start by finding the derivative of f(x) with respect to x:
f'(x) = d/dx [x^2 + (1000 - x)^2]
To simplify, let's break the function down into two parts and find the derivative separately:
f'(x) = d/dx [x^2] + d/dx [(1000 - x)^2]
The derivative of x^2 is 2x, and the derivative of (1000 - x)^2 is 2(1000 - x) * (-1) = -2(1000 - x).
Now, let's combine the two derivatives:
f'(x) = 2x - 2(1000 - x)
Simplifying further:
f'(x) = 2x - 2000 + 2x
f'(x) = 4x - 2000
Now, to find where the function is increasing, we need to determine the values of x for which f'(x) > 0.
4x - 2000 > 0
Adding 2000 to both sides:
4x > 2000
Dividing by 4:
x > 500
From this, we can conclude that the function f(x) = x^2 + (1000 - x)^2 is increasing for x > 500 since f'(x) > 0 in that range.
Now, let's use this fact to determine which number is larger: 1000^2 or 998^2 + 2^2.
Notice that 1000 is greater than 998 + 2. As the function f(x) is increasing for x > 500, it means that the larger the value of x, the greater the value of f(x). Since 1000 > 500, we can assume that f(1000) is greater than f(998 + 2).
Let's calculate f(1000) and f(998 + 2) to confirm this:
f(1000) = 1000^2 + (1000 - 1000)^2 = 1000000 + 0 = 1000000
f(998 + 2) = (998 + 2)^2 + (1000 - (998 + 2))^2 = 1000000 + 0 = 1000000
Both f(1000) and f(998 + 2) are equal to 1000000, which means that the values are the same.
Therefore, the answer is that a) 1000^2 and b) 998^2 + 2^2 are equal.