Question

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ f(x)=\frac{64}{\sin x}+\frac{27}{\cos x} \text { on }(0, \pi / 2) $$

Short answer

The function has a global minimum at approximately \( x = 0.846 \), but no global maximum within the interval.

Step-by-step solution

Identify the Function and Interval

We need to find the global maximum and minimum for the function \( f(x) = \frac{64}{\sin x} + \frac{27}{\cos x} \) on the interval \((0, \frac{\pi}{2})\). The endpoints of the interval are not included, so we focus on where the function might have critical points within the interval.

Find the Derivative of the Function

To find critical points, we first find the derivative of the function. Using the quotient rule, the derivative of \( f(x) = \frac{64}{\sin x} + \frac{27}{\cos x} \) is:\[f'(x) = -64 \frac{\cos x}{\sin^2 x} - 27 \frac{\sin x}{\cos^2 x}\]This derivative must be set to zero to find critical points.

Solve for Critical Points

Set the derivative to zero:\[-64 \frac{\cos x}{\sin^2 x} - 27 \frac{\sin x}{\cos^2 x} = 0\]This simplifies to:\[\frac{64 \cos x}{\sin^2 x} = \frac{27 \sin x}{\cos^2 x}\]Cross-multiply and simplify:\[64 \cos^3 x = 27 \sin^3 x\]Solve this equation to find possible values of \(x\) in the interval \((0, \frac{\pi}{2})\).

Test Critical Points and Interval Ends

The critical point occurs where \( \cos x = (27/64)^{1/3} \sin x \). Solving gives us \( x \approx 0.846 \) as a critical point in \((0, \frac{\pi}{2})\). Evaluate \( f(x) \) at this point and compare it to very near values to 0 and \( \frac{\pi}{2} \), as the function values tend towards infinity at these limits.

Evaluate and Determine Extrema

Calculate \( f(0.846) \) and observe the behavior towards the endpoints near \( x = 0 \) and \( x = \frac{\pi}{2} \). Due to the characteristics of sin and cos, these pressures make \( f(x) \to \infty \) near both ends. By evaluating \( x = 0.846 \), we observe it as a minimum value point. Thus, the function has no global maximum but has a minimum approximately at \( x = 0.846 \) within the interval.

Key concepts

Critical Points

In calculus optimization, critical points are crucial in determining where a function might reach its local maximum or minimum values. A critical point occurs where the derivative of a function is zero or undefined. In the given exercise, the focus is on finding critical points of the function \[ f(x) = \frac{64}{\sin x} + \frac{27}{\cos x} \] within the open interval \((0, \frac{\pi}{2})\).To find these points, we first calculate the derivative and set it equal to zero. By setting the derivative equation \[ -64 \frac{\cos x}{\sin^2 x} - 27 \frac{\sin x}{\cos^2 x} = 0 \] to zero, we identify where the slope of the tangent to the curve is zero, indicating potential critical points. The process requires solving the resulting equation to pinpoint any eligible values of \( x \) within the interval. In this problem, the critical point of approximately \( x = 0.846 \) was found within the desired range.

Derivatives

Derivatives form the backbone of calculus as they describe how a function changes at any given point. By examining the derivative, we can understand the rate of change of the function and the dynamics of its graph. In this exercise, the function \[ f(x) = \frac{64}{\sin x} + \frac{27}{\cos x} \] requires finding its derivative using the quotient rule. The quotient rule is used because both terms are in a fractional form. The rule states that if you have a function \( f(x) = \frac{u(x)}{v(x)} \), its derivative is \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2} \]. Applying this principle correctly helps us determine a formula for the derivative: \[ f'(x) = -64 \frac{\cos x}{\sin^2 x} - 27 \frac{\sin x}{\cos^2 x} \]. Finding where this derivative becomes zero identifies points where the function might change its behavior from increasing to decreasing or vice versa.

Global Maximum and Minimum Values

The global maximum and minimum of a function correspond to the highest and lowest points on its graph within a given interval. For the given function \[ f(x) = \frac{64}{\sin x} + \frac{27}{\cos x} \], the task was to find these values on the interval \((0, \frac{\pi}{2})\).In this exercise, it is crucial first to understand that the interval does not include its endpoints, meaning the function cannot be analyzed exactly at \( x=0 \) or \( x=\frac{\pi}{2} \). However, these points were considered to understand the function's behavior as it approaches them—a technique called endpoint analysis.By evaluating the critical point identified at \( x \approx 0.846 \), the function's value is compared to its asymptotic tendencies at the interval's boundaries. The nature of sine and cosine functions causes the function to approach infinity at these ends, suggesting that while no global maximum exists, a minimum value does appear close to \( x = 0.846 \).

Interval Analysis

Interval analysis focuses on examining a function's behavior within a given range. The interval in this instance is \((0, \frac{\pi}{2})\), where the endpoints are not considered part of the function's domain.The importance of interval analysis lies in carefully scrutinizing how the function behaves near these edges. The function at both ends of the interval approaches infinity due to the nature of the sine and cosine in the denominators, providing insight into the boundary behavior and helping to identify any extreme points within the range. Checking nearby values to the endpoints also helps verify the critical points' validity found within the interval—such as the minimum established around \( x = 0.846 \). This type of analysis shows that the function tends to rapidly increase near the limits due to division by values approaching zero in the sine and cosine functions, aiding in understanding the function's entire behavior.