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Find the points of local maxima or minima for the function f(x) = x3.ex.
- a)x=-3 is a point of local maxima
- b)x=-3 is a point of local minima
- c)x=0 is a point of local maxima
- d)x=0 is a point of local minima
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Find the points of local maxima or minima for the function f(x)...
Solution:
The given function is f(x) = x3.ex.
To find the points of local maxima or minima, we need to find the critical points of the function.
Critical points: The points where the derivative of the function is either zero or does not exist.
f'(x) = 3x2.ex + x3.ex
Let f'(x) = 0, then
3x2.ex + x3.ex = 0
x2(ex + x) = 0
x = 0 or x = -ex
Now, we need to check the nature of critical points using the second derivative test.
f''(x) = 6x.ex + 6x2.ex + 2x3.ex
At x = 0,
f''(0) = 0
Thus, x = 0 is not a point of local maxima or minima.
At x = -ex,
f''(-ex) = 6(-ex).ex + 6(-ex)2.ex + 2(-ex)3.ex
f''(-ex) = -2ex3 < />
Thus, x = -ex is a point of local maxima.
Hence, option B is the correct answer.
Note: The second derivative test is used to determine the nature of critical points. If f''(x) > 0, then the critical point is a point of local minima. If f''(x) < 0,="" then="" the="" critical="" point="" is="" a="" point="" of="" local="" maxima.="" if="" f''(x)="0," then="" the="" test="" is="" inconclusive.="" 0,="" then="" the="" critical="" point="" is="" a="" point="" of="" local="" maxima.="" if="" f''(x)="0," then="" the="" test="" is="" />
The given function is f(x) = x3.ex.
To find the points of local maxima or minima, we need to find the critical points of the function.
Critical points: The points where the derivative of the function is either zero or does not exist.
f'(x) = 3x2.ex + x3.ex
Let f'(x) = 0, then
3x2.ex + x3.ex = 0
x2(ex + x) = 0
x = 0 or x = -ex
Now, we need to check the nature of critical points using the second derivative test.
f''(x) = 6x.ex + 6x2.ex + 2x3.ex
At x = 0,
f''(0) = 0
Thus, x = 0 is not a point of local maxima or minima.
At x = -ex,
f''(-ex) = 6(-ex).ex + 6(-ex)2.ex + 2(-ex)3.ex
f''(-ex) = -2ex3 < />
Thus, x = -ex is a point of local maxima.
Hence, option B is the correct answer.
Note: The second derivative test is used to determine the nature of critical points. If f''(x) > 0, then the critical point is a point of local minima. If f''(x) < 0,="" then="" the="" critical="" point="" is="" a="" point="" of="" local="" maxima.="" if="" f''(x)="0," then="" the="" test="" is="" inconclusive.="" 0,="" then="" the="" critical="" point="" is="" a="" point="" of="" local="" maxima.="" if="" f''(x)="0," then="" the="" test="" is="" />
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Find the points of local maxima or minima for the function f(x) = x3.ex.a)x=-3 is a point of local maximab)x=-3 is a point of local minimac)x=0 is a point of local maximad)x=0 is a point of local minimaCorrect answer is option 'B'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Find the points of local maxima or minima for the function f(x) = x3.ex.a)x=-3 is a point of local maximab)x=-3 is a point of local minimac)x=0 is a point of local maximad)x=0 is a point of local minimaCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the points of local maxima or minima for the function f(x) = x3.ex.a)x=-3 is a point of local maximab)x=-3 is a point of local minimac)x=0 is a point of local maximad)x=0 is a point of local minimaCorrect answer is option 'B'. Can you explain this answer?.
Find the points of local maxima or minima for the function f(x) = x3.ex.a)x=-3 is a point of local maximab)x=-3 is a point of local minimac)x=0 is a point of local maximad)x=0 is a point of local minimaCorrect answer is option 'B'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Find the points of local maxima or minima for the function f(x) = x3.ex.a)x=-3 is a point of local maximab)x=-3 is a point of local minimac)x=0 is a point of local maximad)x=0 is a point of local minimaCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the points of local maxima or minima for the function f(x) = x3.ex.a)x=-3 is a point of local maximab)x=-3 is a point of local minimac)x=0 is a point of local maximad)x=0 is a point of local minimaCorrect answer is option 'B'. Can you explain this answer?.
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