91Ó°ÊÓ

To find the maximum value of the expression \( \left|\frac{c^2 - c}{\cos c}\right| \) in the given interval \([0, \frac{\pi}{4}]\), we first need to break down its components. The expression consists of a numerator, \( c^2 - c \), and a denominator, \( \cos c \). To understand the behavior of the entire expression, it is crucial to analyze how each part behaves on its own.

The numerator \( c^2 - c \) is a quadratic function and the denominator \( \cos c \) is a trigonometric function. Both are affected by the interval in which \( c \) lies. By examining each part separately, we can understand how the expression as a whole behaves within the specified range.

The numerator of our expression, \( c^2 - c \), represents a quadratic function. Quadratics are polynomial functions of degree two, characterized by their parabolic shapes. This particular quadratic can be factored into \( c(c-1) \). This means:

• When \( c = 0 \), \( c^2 - c = 0 \).
• When \( c = 1 \), \( c^2 - c = 0 \) as well.

Between 0 and 1, the function is negative, and beyond 1, it becomes positive. However, since \( c \) is restricted to the range \([0, \frac{\pi}{4}]\), which is approximately \( [0, 0.785] \), we are primarily interested in how it behaves between 0 and \( \frac{\pi}{4} \).

The vertex form of the quadratic gives insight into its maximum or minimum. In our interval, \( c(c-1) \) is decreasing at first, reaches zero, and then increases. Within \([0, \frac{\pi}{4}]\), the maximum is achieved at the right end, at \( c = \frac{\pi}{4} \), where \( c^2 - c \) attains approximately 0.61685, as calculated from \(\left(\frac{\pi}{4}\right)^2 - \frac{\pi}{4} \).

The denominator, \( \cos c \), is a trigonometric function, specifically the cosine function, which is well known for its oscillatory behavior. In the interval \([0, \frac{\pi}{4}]\), the cosine function decreases from 1 to roughly 0.707. Here are some key points:

• The cosine of 0 is 1, meaning the function starts at its maximum value.
• At \( c = \frac{\pi}{4} \), \( \cos c = \frac{1}{\sqrt{2}} \approx 0.707 \).

This decrease impacts the expression as the denominator's value getting smaller increases the expression's magnitude for positive numerators. Thus, the smallest value it takes directly results in the largest value of the expression, due to division.
Understanding this behavior allows us to use this minimal value to evaluate its impact on the whole expression.

Evaluating the interval \([0, \frac{\pi}{4}]\) is crucial for determining the maximum value of the expression. This step involves summarizing how the individual components - the quadratic and trigonometric functions - behave within this range.

By determining the maximum value of the quadratic within this interval, which is \(\left(\frac{\pi}{4}\right)^2 - \frac{\pi}{4}\) at \( c = \frac{\pi}{4} \), and knowing the minimum value of the cosine, which is \(\frac{1}{\sqrt{2}}\) also at \( c = \frac{\pi}{4} \), we can find the maximum bound of the whole expression.

• Dividing the maximum numerator by the minimum denominator gives: \[\frac{(\frac{\pi}{4})^2 - \frac{\pi}{4}}{\frac{1}{\sqrt{2}}}\]
• This equates to approximately \( 0.238 \).

The interval evaluation thus culminates in a bound that offers a precise estimation of the potential maximum value of the expression within the given domain.