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Chapter 7: Problem 74

Assertion \((\mathbf{A}):\) Let \(f(x)=5-4(x-2)^{21}\), then at \(x=2\) the function \(f(x)\) attains neither the least value nor the greatest value. Reason \((\mathbf{R}):\) At \(x=2\), the first derivative does not exist.

Short Answer

Expert verified
Explain your answer. Answer: Yes, at x=2, the function f(x) attains neither the least value nor the greatest value. Although the first derivative exists at x=2, it is equal to 0, making the point a critical point. The second derivative test is inconclusive, as the second derivative also equals 0 at x=2. The function itself is an odd degree polynomial with a negative leading coefficient, which indicates that it does not have a global maximum or minimum value at x=2. The first derivative does not provide information about local maximum or minimum values, so we can conclude that the function attains neither the least value nor the greatest value at x=2.

Step by step solution

01

Determine the first derivative of the function f(x)

To begin with, we need to find the first derivative of the given function, \(f(x) = 5 - 4(x-2)^{21}\). Using the chain rule, we have: \(f'(x) = -4 \cdot 21 (x - 2)^{20}\) Now we can analyze the first derivative at \(x=2\).
02

Evaluate the first derivative at x=2

Substitute \(x=2\) into the first derivative: \(f'(2) = -4 \cdot 21 (2 - 2)^{20} = -84(0)^{20} = 0\) Since the first derivative exists and is equal to 0 at \(x=2\), we can say that this point is a critical point.
03

Determine if the given point is a local minimum, local maximum, or neither.

Since the first derivative exists at \(x=2\), then we can use the second derivative test to determine if the function attains a minimum or a maximum value at this point. Find the second derivative of the function: \(f''(x) = -84 \cdot 20(x - 2)^{19}\) Now, evaluate the second derivative at x=2: \(f''(2) = -84 \cdot 20 (2 - 2)^{19} = -1680(0)^{19} = 0\) Since the second derivative is also 0 at x=2, the second derivative test is inconclusive. We need to look at the original function to determine the behavior of the function around x=2.
04

Analyze the behavior of the original function around x=2

Looking at the original function \(f(x) = 5 - 4(x-2)^{21}\), we notice that it is an odd degree polynomial with a negative leading coefficient. This means that as x approaches \(-\infty\), the function approaches \(+\infty\) and as x approaches \(+\infty\), the function approaches \(-\infty\). So, the function will not attain a global maximum or minimum value at \(x=2\). As the first derivative does not give any information about local maximum or minimum as well, we can conclude that at \(x=2\) the function \(f(x)\) attains neither the least value nor the greatest value. This confirms the assertion \((\mathbf{A})\). The reason \((\mathbf{R})\) given is incorrect, as the first derivative does exist at \(x=2\).

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