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The function f(x) = 8loge x - x2 + 3 attains its minimum over the interval [1, e] at x = (Here loge x is the natural logarithm of x )
- a)e
- b)2
- c)(1 + e)/2
- d)1
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The function f(x) = 8loge x - x2 + 3 attains its minimum over the int...
Explanation:
To find the minimum of the function f(x) = 8loge x - x^2 over the interval [1, e], we need to find the critical points of the function and evaluate the function at those points to determine the minimum.
Finding the Critical Points:
To find the critical points, we need to find the derivative of the function f(x) and set it equal to zero.
f(x) = 8loge x - x^2
Differentiating f(x) with respect to x, we get:
f'(x) = 8(1/x) - 2x
Setting f'(x) = 0 and solving for x:
8(1/x) - 2x = 0
Multiplying through by x, we get:
8 - 2x^2 = 0
Rearranging the equation:
2x^2 = 8
x^2 = 4
Taking the square root of both sides:
x = ±2
Evaluating the Function:
Now we need to evaluate the function f(x) at the critical points and at the endpoints of the given interval [1, e].
f(1) = 8loge(1) - 1^2 = 0 - 1 = -1
f(e) = 8loge(e) - e^2 = 8(1) - e^2 = 8 - e^2
f(2) = 8loge(2) - 2^2 = 8(0.693) - 4 = 5.544 - 4 = 1.544
So, we have the following values:
f(1) = -1
f(e) = 8 - e^2
f(2) = 1.544
Comparing the Values:
To determine the minimum value of the function, we compare the values of f(x) at the critical points and endpoints.
f(1) = -1 is the smallest value among all the values obtained.
Therefore, the minimum value of the function f(x) over the interval [1, e] is at x = 1.
Hence, the correct answer is option D) 1.
To find the minimum of the function f(x) = 8loge x - x^2 over the interval [1, e], we need to find the critical points of the function and evaluate the function at those points to determine the minimum.
Finding the Critical Points:
To find the critical points, we need to find the derivative of the function f(x) and set it equal to zero.
f(x) = 8loge x - x^2
Differentiating f(x) with respect to x, we get:
f'(x) = 8(1/x) - 2x
Setting f'(x) = 0 and solving for x:
8(1/x) - 2x = 0
Multiplying through by x, we get:
8 - 2x^2 = 0
Rearranging the equation:
2x^2 = 8
x^2 = 4
Taking the square root of both sides:
x = ±2
Evaluating the Function:
Now we need to evaluate the function f(x) at the critical points and at the endpoints of the given interval [1, e].
f(1) = 8loge(1) - 1^2 = 0 - 1 = -1
f(e) = 8loge(e) - e^2 = 8(1) - e^2 = 8 - e^2
f(2) = 8loge(2) - 2^2 = 8(0.693) - 4 = 5.544 - 4 = 1.544
So, we have the following values:
f(1) = -1
f(e) = 8 - e^2
f(2) = 1.544
Comparing the Values:
To determine the minimum value of the function, we compare the values of f(x) at the critical points and endpoints.
f(1) = -1 is the smallest value among all the values obtained.
Therefore, the minimum value of the function f(x) over the interval [1, e] is at x = 1.
Hence, the correct answer is option D) 1.
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Community Answer
The function f(x) = 8loge x - x2 + 3 attains its minimum over the int...
Method 1
Given: Function f(x) = 8loge x - x2 + 3
Differentiate above function with respect to x,
For maxima or minima,
f '(x) = 0
8/x = 2x
x2 = 4
x = ± 2
So, we take x = 2 ∈[1,e]
f "(2) < />
At x = 2 , function f (x) is maximum. Therefore, function f(x) is attains minimum in closed intervals [1, e] .
f(x)min = min { f (1), f (e)}
At x = 1 , f(1) = 8loge 1 - 1 + 3 = 0 - 1 + 3 = 2
At x =e , f(e) = 8loge e - e2 + 3
f (e) = 8 + 3 - e2
f(e) = 11 - 7.38 = 3.61
f (x)min = min{2, 3.61} = 2
We can see, that minimum value occur at point x = 1 .
Hence, the correct option is (D).
Method 2
Given function, f(x) = 8loge x - x2 + 3, x ∈ [1,e] as closed interval is given.
Checking options,
Option (A) : x = e,
f(x = e) = 8loge e - e2 + 3
f(x = e) = 8 - e2 +3 = 3.61
Option (B) : x = 2,
f (x = 2) = 8loge (2) - 22 + 3
f(x = 2) = 4.545
Option (C) : 
Option (D) : x = 1,
f(x = 1) = 8loge(1) - (1)2 + 3
f(x =1) = 8 x 0 -1+ 3 = 2
At x = 1 it shows minimum value.
Hence, the correct option is (D).
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The function f(x) = 8loge x - x2 + 3 attains its minimum over the interval [1, e] at x = (Here loge x is the natural logarithm of x )a)eb)2c)(1 + e)/2d)1Correct answer is option 'D'. Can you explain this answer?
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