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If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R?
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If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other,...
Problem:
Find the maximum value of the function f(x), where x belongs to R, if one root of the equation t^2 - (12x)t - f(x) 64x=0 is twice of other.
Solution:
Let the roots of the equation be a and 2a, then we have:
t^2 - (12x)t - f(x) 64x = 0
(t - a)(t - 2a) = 0
t^2 - (3a)t + 2a^2 = 0
Comparing the coefficients of t in both equations, we get:
3a = 12x
a = 4x
Substituting the value of a in the second equation, we get:
16x^2 - 12x(4x) + 2a^2 = 0
16x^2 - 48x^2 + 2a^2 = 0
2a^2 = 32x^2
Therefore, the two roots of the equation are:
a = 4x
2a = 8x
Sum of the roots = a + 2a = 12x
Using sum and product of roots formula, we get:
a + 2a = 12x = -b/a
a(2a) = 8ax^2 = c/a
Substituting the value of a in c/a, we get:
c/a = f(x)64x
8x^2 = f(x)64x
f(x) = x/8
Therefore, the maximum value of f(x) is 0.125, which occurs when x = 1.
Answer:
The maximum value of the function f(x) is 0.125, which occurs when x = 1.
Find the maximum value of the function f(x), where x belongs to R, if one root of the equation t^2 - (12x)t - f(x) 64x=0 is twice of other.
Solution:
Let the roots of the equation be a and 2a, then we have:
t^2 - (12x)t - f(x) 64x = 0
(t - a)(t - 2a) = 0
t^2 - (3a)t + 2a^2 = 0
Comparing the coefficients of t in both equations, we get:
3a = 12x
a = 4x
Substituting the value of a in the second equation, we get:
16x^2 - 12x(4x) + 2a^2 = 0
16x^2 - 48x^2 + 2a^2 = 0
2a^2 = 32x^2
Therefore, the two roots of the equation are:
a = 4x
2a = 8x
Sum of the roots = a + 2a = 12x
Using sum and product of roots formula, we get:
a + 2a = 12x = -b/a
a(2a) = 8ax^2 = c/a
Substituting the value of a in c/a, we get:
c/a = f(x)64x
8x^2 = f(x)64x
f(x) = x/8
Therefore, the maximum value of f(x) is 0.125, which occurs when x = 1.
Answer:
The maximum value of the function f(x) is 0.125, which occurs when x = 1.
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If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R?
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If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R?.
If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If one root of the equation t^2-(12x)t-(f(x) 64x)=0 is twice of other, then find the maximum value of the function f(x),where x belong to R?.
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