Question

Multiple Choice A continuous function \(f\) has domain \([1,25]\) and range \([3,30] .\) If \(f^{\prime}(x)<0\) 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

Short answer

The value of \(f(25)\) is 3 (Option b)

Step-by-step solution

Understanding the given function

A continuous function \(f\) is given with domain \([1,25]\) and range \([3,30]\). This implies that the function takes all values between 3 and 30 for all inputs between 1 and 25. Moreover, \(f'(x)<0\) for all \(x\) in \([1, 25]\) suggests that the function is decreasing over that interval.

Applying the properties of the function

Let's denote \(f(1)=y_1\) and \(f(25)=y_2\). Since \(f'(x)<0\) for all \(x\) in \([1, 25]\), the function \(f(x)\) is decreasing. Hence, for any two points \(x_1\) and \(x_2\) in \([1, 25]\) if \(x_2>x_1\) then \(f(x_2)<f(x_1)\). By applying that to the given domain, we get \(f(25)<f(1)\). Moreover, since \(f(1)\) and \(f(25)\) must lie within the range of the function which is \([3,30]\), \(f(1)\) must be at the highest extreme of the range and \(f(25)\) at the lower extreme of the range for the function to be decreasing in the domain \([1,25]\).

Determine \(f(25)\)

As \(f(1)\) must represent the higher extreme of the range and \(f(25)\) the lower one due to the function being decreasing in the domain, we have \(f(1)=30\) and \(f(25)=3\).

Key concepts

Decreasing Function

A decreasing function is one where the output value decreases as the input value increases. In simpler terms, as you move along the graph from left to right, the curve falls. The condition given in the exercise, where the derivative of the function, denoted by \(f'(x)\), is less than zero \(f'(x)<0\) for all \(x\) between 1 and 25 indicates a decreasing function.

In the context of the problem provided, this property is key to understanding how the function behaves across its domain, leading to the correct insight that the value of \(f(25)\) must be less than the value of \(f(1)\).

For a continuous decreasing function with a closed interval domain like \[1, 25\], the function's value at the start of the interval will be the maximum, and the value at the end of the interval will be the minimum. In this case, since we are given the range as well, it allows us to determine specific values at these points.

Function Domain and Range

The domain of a function represents all the possible input values for which the function is defined, whereas the range represents all the possible output values the function can produce. For the continuous function \(f\) in our exercise, the domain is the closed interval \[1, 25\] and the range is \[3, 30\].

The importance of domain and range in solving calculus problems cannot be overstated, as they set the boundaries for function behavior. In the step-by-step solution, knowing that \(f(1)\) and \(f(25)\) must fall within the given range allows us to conclude that they must correspond to the extreme values of the range, due to the function's decreasing nature. It's not enough to just know that \(f(25)\) should be smaller than \(f(1)\); we need this range to conclude that \(f(25)=3\) and \(f(1)=30\).

Derivative

The derivative of a function measures how the function's output value changes concerning a change in the input value. It is the fundamental tool in calculus used to determine the rate of change of a function. For instance, in our original example, the negative derivative (\(f'(x)<0\)) over an interval implies that the function is decreasing over that interval.

This concept plays a crucial role in understanding the nature of the function's graph, and subsequently, in determining the behavior of the function over its domain. Knowing that the derivative is negative throughout indicates that there are no turning points or intervals where the function might increase within the domain, which is vital information that supports the deduction made in the solution. As the derivative remains less than zero, we can be confident in the function's consistently decreasing behavior, leading to the conclusion that \(f(25)\) must be at the lower end of the function's range.