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F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24?
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F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critica...
Question: Determine the critical points and nature of extremum for the function f(x) = (x - 2) ^ 17 * (x + 5) ^ 24.
Solution:
To find the critical points of f(x), we need to find the values of x where the derivative of f(x) is equal to zero or undefined.
f'(x) = 17(x - 2) ^ 16 * (x + 5) ^ 24 + 24(x - 2) ^ 17 * (x + 5) ^ 23
We can simplify this expression by factoring out (x - 2) ^ 16 * (x + 5) ^ 23:
f'(x) = (x - 2) ^ 16 * (x + 5) ^ 23 * [17(x + 5) + 24(x - 2)]
Setting f'(x) equal to zero and solving for x, we get:
17(x + 5) + 24(x - 2) = 0
41x - 23 = 0
x = 23/41
So, the only critical point of f(x) is x = 23/41.
To determine the nature of extremum at x = 23/41, we need to analyze the sign of f'(x) around this point.
When x < 23/41,="" both="" factors="" (x="" -="" 2)="" ^="" 16="" and="" (x="" +="" 5)="" ^="" 23="" are="" negative.="" so,="" f'(x)="" is="" />
When x > 23/41, both factors (x - 2) ^ 16 and (x + 5) ^ 23 are positive. So, f'(x) is positive.
Therefore, we can conclude that f(x) has a minimum at x = 23/41.
Answer: The correct option is (B) f has a minimum at x = 2.
Solution:
Finding Critical Points
To find the critical points of f(x), we need to find the values of x where the derivative of f(x) is equal to zero or undefined.
f'(x) = 17(x - 2) ^ 16 * (x + 5) ^ 24 + 24(x - 2) ^ 17 * (x + 5) ^ 23
We can simplify this expression by factoring out (x - 2) ^ 16 * (x + 5) ^ 23:
f'(x) = (x - 2) ^ 16 * (x + 5) ^ 23 * [17(x + 5) + 24(x - 2)]
Setting f'(x) equal to zero and solving for x, we get:
17(x + 5) + 24(x - 2) = 0
41x - 23 = 0
x = 23/41
So, the only critical point of f(x) is x = 23/41.
Determining the Nature of Extremum
To determine the nature of extremum at x = 23/41, we need to analyze the sign of f'(x) around this point.
When x < 23/41,="" both="" factors="" (x="" -="" 2)="" ^="" 16="" and="" (x="" +="" 5)="" ^="" 23="" are="" negative.="" so,="" f'(x)="" is="" />
When x > 23/41, both factors (x - 2) ^ 16 and (x + 5) ^ 23 are positive. So, f'(x) is positive.
Therefore, we can conclude that f(x) has a minimum at x = 23/41.
Answer: The correct option is (B) f has a minimum at x = 2.
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F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24?
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F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24?.
F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for F(x) = (x - 2) ^ 17 * (x + 5) ^ 24 Then (A) f does not have a critical point at x = 2 (B) f has a minimum at x = 2 (C) f has neither a maximum nor a minimum at x = 2 (D) f has a maximum at x = 2 মনে কর f(x) = (x - 2) ^ 17 * (x + 5) ^ 24?.
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