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25 Jul 2019, 23:40
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
56% (01:08) correct
44%
(01:11)
wrong
based on 671
sessions
History
Date
Time
Result
Not Attempted Yet
A function is defined by \(f(x) = x^2 + 4x – 5\). What is the minimum value of \(f(x)\)?
(A) 1
(B) 0
(C) –5
(D) –8
(E) –9
(A) 1
(B) 0
(C) –5
(D) –8
(E) –9
26 Jul 2019, 01:15
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The function defined in the question is a quadratic function with a positive co-efficient for \(x^2\). There are two methods which can be adopted to solve this question.
Let’s discuss both of them in detail.
For a quadratic function of the form a\(x^2\) + bx + c where a>0, the minimum value of the function is given by \(\frac{4ac – b^2}{4a}\).
Comparing the function given in the question with the standard form, we have,
a = 1, b = 4 and c = -5.
Substituting these values in the expression above, we have,
Minimum value of f(x) = \(\frac{4 * 1 * (-5) – (4)^2}{4*1}\) = \(\frac{- 20 – 16}{4}\) = -9.
For a quadratic function of the form a\(x^2\) + bx + c where a>0, the minimum value of the function occurs at x = \(\frac{-b}{2a}\).
This means, for the given function, the minimum value of the function occurs at x = \(\frac{-4}{2}\) = -2.
Substituting this value in the function, we have f(-2) = 4 – 8 – 5 = -9.
Therefore, the minimum value of the given function is -9. The correct answer option is E.
Hope this helps!
Let’s discuss both of them in detail.
For a quadratic function of the form a\(x^2\) + bx + c where a>0, the minimum value of the function is given by \(\frac{4ac – b^2}{4a}\).
Comparing the function given in the question with the standard form, we have,
a = 1, b = 4 and c = -5.
Substituting these values in the expression above, we have,
Minimum value of f(x) = \(\frac{4 * 1 * (-5) – (4)^2}{4*1}\) = \(\frac{- 20 – 16}{4}\) = -9.
For a quadratic function of the form a\(x^2\) + bx + c where a>0, the minimum value of the function occurs at x = \(\frac{-b}{2a}\).
This means, for the given function, the minimum value of the function occurs at x = \(\frac{-4}{2}\) = -2.
Substituting this value in the function, we have f(-2) = 4 – 8 – 5 = -9.
Therefore, the minimum value of the given function is -9. The correct answer option is E.
Hope this helps!
26 Jul 2019, 01:10
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Solution
Given:
- • \(f(x) = x^2 + 4x – 5\)
To find:
- • The minimum value of f(x)
Approach and Working Out:
- • f(x) = \(x^2 + 4x – 5 = (x + 2)^2 – 9\)
• The minimum value of \((x + 2)^2 = 0\)
• Therefore the minimum value of f(x) = 0 – 9 = -9
Hence, the correct answer is Option E.
Answer: E
