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If minimum value of f(x) = x2 + 2bx + 2c2 is greater than maximum value of g(x) = - x2 - 2cx + b2 then for x is real
- a)∣c∣ > ∣b∣ √2
- b)∣c∣ √2 > b
- c)0 < c < √2b
- d)No real value of a
Correct answer is option 'A'. Can you explain this answer?
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If minimum value of f(x) = x2 + 2bx + 2c2 is greater than maximum valu...
Understanding the Problem:
The problem involves comparing the minimum value of a quadratic function f(x) with the maximum value of another quadratic function g(x).
Given:
f(x) = x^2 + 2bx + 2c^2
g(x) = -x^2 - 2cx + b^2
Analysis:
To find the minimum value of f(x) and the maximum value of g(x), we need to consider the vertex of the parabolas formed by the quadratic functions.
Minimum value of f(x):
The minimum value of f(x) occurs at the vertex of the parabola.
The x-coordinate of the vertex is given by -b/2a.
Substitute this x-value into f(x) to find the minimum value.
Maximum value of g(x):
The maximum value of g(x) occurs at the vertex of the parabola.
The x-coordinate of the vertex is given by -b/2a.
Substitute this x-value into g(x) to find the maximum value.
Comparison:
If the minimum value of f(x) is greater than the maximum value of g(x), then we have:
2c^2 > b^2
Taking the square root of both sides gives:
√2c > b
Therefore, the correct answer is option 'A': c > √2b.
The problem involves comparing the minimum value of a quadratic function f(x) with the maximum value of another quadratic function g(x).
Given:
f(x) = x^2 + 2bx + 2c^2
g(x) = -x^2 - 2cx + b^2
Analysis:
To find the minimum value of f(x) and the maximum value of g(x), we need to consider the vertex of the parabolas formed by the quadratic functions.
Minimum value of f(x):
The minimum value of f(x) occurs at the vertex of the parabola.
The x-coordinate of the vertex is given by -b/2a.
Substitute this x-value into f(x) to find the minimum value.
Maximum value of g(x):
The maximum value of g(x) occurs at the vertex of the parabola.
The x-coordinate of the vertex is given by -b/2a.
Substitute this x-value into g(x) to find the maximum value.
Comparison:
If the minimum value of f(x) is greater than the maximum value of g(x), then we have:
2c^2 > b^2
Taking the square root of both sides gives:
√2c > b
Therefore, the correct answer is option 'A': c > √2b.
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If minimum value of f(x) = x2 + 2bx + 2c2 is greater than maximum value of g(x) = - x2 - 2cx + b2 then for x is reala)c > b √2b)c √2 > bc)0 < c < √2bd)No real value of aCorrect answer is option 'A'. Can you explain this answer?
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