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14 Jul 2024, 07:00
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C
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Difficulty:
65%
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Question Stats:
57% (02:46) correct
43%
(02:27)
wrong
based on 86
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F(x) is a quadratic equation having value 4 at x=0 . The minimum value of F(x) occurs at x=2. Find the value of F(8) if the minimum value of F(x) is 0.
A. 18
B. 42
C. 36
D. 30
E. 24
A. 18
B. 42
C. 36
D. 30
E. 24
DG1989
Joined: 16 Feb 2023
Last visit: 24 Dec 2024
Posts: 140
Own Kudos:
Given Kudos: 9
Location: India
Concentration: Finance, Technology
Schools: Kellogg '26
GPA: 4
14 Jul 2024, 11:37
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F(x) is a quadratic equation having value 4 at x=0. The minimum value of F(x) occurs at x=2.
Find the value of F(8) if the minimum value of F(x) is 0.
A. 18
B. 42
C. 36
D. 30
E. 24
Solution: Let the quadratic equation F(x) = a\(x^2\) + bx + c
Given that F(0) = 4
This means, a\((0)^2\) + b(0) + c = 4
c = 4
Thus F(x) = a\(x^2\) + bx + 4 ----------(1)
It is also given that the minimum value of F(x) occurs at x = 2 and is equal to 0
This means, F(2) = 0
Substituting x = 2 in (1)
4a + 2b + 4 = 0 ----------(2)
To find the value at which the minimum value of F(x) occurs we equate its derivative to 0 i.e.,
2ax + b = 0
But minimum value of F(x) occurs at x = 2
Hence
4a + b = 0 ----------(3)
Using (3) in (2)
4a + b + b + 4 = 0
b + 4 = 0
b = -4
substituting the value of b in (3)
4a - 4 = 0
a = 1
We need to find F(8)
Substituting x = 8 and the values of a and b in (1)
F(8) = 64a + 8b + 4
= 64 - 32 + 4
= 36
Hence, C is the correct answer
Find the value of F(8) if the minimum value of F(x) is 0.
A. 18
B. 42
C. 36
D. 30
E. 24
Solution: Let the quadratic equation F(x) = a\(x^2\) + bx + c
Given that F(0) = 4
This means, a\((0)^2\) + b(0) + c = 4
c = 4
Thus F(x) = a\(x^2\) + bx + 4 ----------(1)
It is also given that the minimum value of F(x) occurs at x = 2 and is equal to 0
This means, F(2) = 0
Substituting x = 2 in (1)
4a + 2b + 4 = 0 ----------(2)
To find the value at which the minimum value of F(x) occurs we equate its derivative to 0 i.e.,
2ax + b = 0
But minimum value of F(x) occurs at x = 2
Hence
4a + b = 0 ----------(3)
Using (3) in (2)
4a + b + b + 4 = 0
b + 4 = 0
b = -4
substituting the value of b in (3)
4a - 4 = 0
a = 1
We need to find F(8)
Substituting x = 8 and the values of a and b in (1)
F(8) = 64a + 8b + 4
= 64 - 32 + 4
= 36
Hence, C is the correct answer
14 Jul 2024, 14:41
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Harsha
Enthu about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com
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Harsha
Enthu about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com
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