Question

Find the extreme values of the function on the given interval. \(f(x)=e^{x} \cos x\) on \([0, \pi]\)

Short answer

The minimum value of \(f(x)\) on \([0, \pi]\) is \(-e^{\pi}\) at \(x = \pi\), and the maximum is \(\frac{e^{\frac{\pi}{4}}}{\sqrt{2}}\) at \(x=\frac{\pi}{4}\).

Step-by-step solution

Find the Derivative

To find the extreme values of the function, we first need to find its derivative. For the function \(f(x) = e^x \cos x\), we apply the product rule, which states that \((uv)' = u'v + uv'\). Here, \(u = e^x\) and \(v = \cos x\). So, we have:\[f'(x) = (e^x)' \cdot \cos x + e^x \cdot (\cos x)'\]Calculate the derivatives:\( (e^x)' = e^x \) and \( (\cos x)' = -\sin x \).Thus, \[f'(x) = e^x \cos x - e^x \sin x = e^x(\cos x - \sin x)\]This derivative must be evaluated to find critical points.

Solve for Critical Points

Set the derivative equal to zero to find the critical points:\[f'(x) = e^x(\cos x - \sin x) = 0\]Since \(e^x eq 0\) for any \(x\), it implies:\[\cos x - \sin x = 0\]So, \(\cos x = \sin x\). Solving for \(x\), you get \(x = \frac{\pi}{4}\) in the interval \([0, \pi]\).

Evaluate the Function at Critical Points and Endpoints

Next, we must evaluate \(f(x)\) at the critical point and the endpoints of the interval. Calculate \(f(x) = e^x \cos x\) at \(x = 0, \frac{\pi}{4}, \pi \).- \(f(0) = e^0 \cdot \cos 0 = 1 \cdot 1 = 1\)- \(f\left(\frac{\pi}{4}\right) = e^{\frac{\pi}{4}} \cdot \cos\left(\frac{\pi}{4}\right) = \frac{e^{\frac{\pi}{4}}}{\sqrt{2}}\)- \(f(\pi) = e^{\pi} \cdot (-1)\) since \(\cos \pi = -1\). Thus, \(f(\pi) = -e^{\pi}\).

Determine Extreme Values

By comparing the values, we can identify the extreme values:- At \(x = 0\), \(f(x) = 1\)- At \(x = \frac{\pi}{4}\), \(f(x) = \frac{e^{\frac{\pi}{4}}}{\sqrt{2}}\) (approximately 1.285)- At \(x = \pi\), \(f(x) = -e^{\pi}\) (which is a large negative number)The minimum value is \(-e^{\pi}\) at \(x = \pi\) and the maximum value is attained at \(x = \frac{\pi}{4}\).

Key concepts

Extreme Values

Extreme values in calculus are those points on the graph of a function where the function reaches its highest or lowest points, known as maxima or minima. These values are often found on a specified interval. In the context of our exercise, we are tasked with finding these extreme values within the interval \([0, \pi]\). This means we are considering the behavior of the function \(f(x) = e^x \cos x\) only between \([0, \pi]\), and not outside these bounds.
The strategy to find extreme values is to identify critical points—these are the points where the derivative of the function is zero or undefined. Additionally, we also need to evaluate the function at the endpoints of the given interval, as extreme values can occur there as well.
In step-by-step solutions, once we have our critical points and endpoint evaluations, we compare the function's values at these crucial spots. This comparison reveals the local or global maximum and minimum values on the interval.

Derivative

The derivative is a fundamental concept in calculus that represents the rate at which a function is changing at any given point. It is a powerful tool for analyzing functions and finding their extreme values. To find the derivative of a product of functions, like in our exercise \(f(x) = e^x \cos x\), we use the product rule.

• The product rule states that for two functions \(u(x)\) and \(v(x)\), \( (uv)' = u'v + uv'\).
• In our specific example, we identified \(u = e^x\) and \(v = \cos x\).
• Upon differentiation, we find that \( (e^x)' = e^x\) and \( (\cos x)' = -\sin x\).

With these, we can construct the derivative:
\[ f'(x) = e^x \cos x + e^x (-\sin x) = e^x (\cos x - \sin x) \]
The derivative \( f'(x) \) helps us pinpoint where the function is increasing or decreasing by setting it to zero and solving for the values of \(x\). These places where the slope is zero are our critical points.

Interval

Intervals in calculus refer to the specified range of \(x\)-values in which we're interested in studying the behavior of a function. In many problems like our current one, intervals are given and they guide us in determining the relevant portion of the function to analyze.
The interval \[0, \pi\]\ given in the exercise defines the section of the x-axis over which the function \(f(x) = e^x \cos x\) is examined. It begins at \(x = 0\) and ends at \(x = \pi\), inclusive.
To evaluate extreme values, we not only look at any critical points found within this interval but also consider the endpoints themselves. This is crucial because the function can achieve its extreme values at these points, as shown by evaluating \( f(0)\) and \( f(\pi)\) in the solution. Understanding intervals aids us in defining limits for our calculations and ensuring we only consider the relevant data when determining maxima and minima.