a continuous function f has domain (1,25) and range (3,30). if f'(x) is less than 0 for all x between 1 and 25, what is f(25)?

Question

A visually appealing image that corresponds to a mathematical concept, specifically dealing with calculus. The image should depict the concept of a continuous function with a defined domain and range. It can show an example of a function that starts high and slopes downward across a plotted graph, as that's what would happen if the derivative of the function is less than 0 for all the values between 1 and 25. However, don't incorporate any text or explicit indicators of specific values into the image.

a continuous function f has domain (1,25) and range (3,30). if f'(x) is less than 0 for all x between 1 and 25, what is f(25)?

a)1
b)3
c)25
d)30
e)impossible to determine from the information given

Answer 1

draw a rectangle with vertices

(1,30), (1,3), (25,3) and (25,30)

notice domain is from 1 to 25
and the range is from 3 to 30

your graph must lie within that rectangle, and it is constantly decreasing. ( f' (x) < 0 )

Since the domain ends at 25 and the range is as low as 3,
(25,3) must be the endpoint of your graph

so f(25) = 3

Answer 2

Well, it seems like our function f is like a sneaky roller coaster that only goes downhill. Since f'(x) is always less than 0 between 1 and 25, that means the slope of the function is negative everywhere. That means the function is decreasing as we move along the x-axis.

So, if f(25) is in the range of (3,30), it must be less than 30 but still greater than 3. Since the function is constantly decreasing, we can safely say that f(25) must be closer to 3 than to 30.

Therefore, the only answer option that fits the bill is b) 3. Oh, how I love it when humor and math come together!

Answer 3

Given that f'(x) is less than 0 for all x between 1 and 25, we can conclude that f(x) is a decreasing function on the interval (1,25).

Since the domain of f is (1,25) and the range of f is (3,30), we know that f(1) is between 3 and 30, and f(25) is also between 3 and 30.

However, since f is a decreasing function, the value of f(25) must be less than the value of f(1). Therefore, f(25) cannot be equal to 30, as option (d) suggests.

Therefore, the correct answer is e) impossible to determine from the information given.

Answer 4

To find the value of f(25), we need to use the information given.

Given that f'(x) is less than 0 for all x between 1 and 25, this means that the derivative of f(x) is negative over the entire interval (1, 25).

Since the derivative represents the rate of change of a function, a negative derivative means that the function is decreasing throughout the interval (1, 25).

Now, because f is a continuous function with a decreasing rate of change, it must approach its minimum value at the upper bound of its domain, which is 25 in this case.

Since the range of f is (3, 30), and the function is decreasing, the minimum value that f(25) can attain is the lower bound of the range, which is 3.

Therefore, f(25) = 3.

The correct answer is b) 3.