Find the absolute maximum and the absolute minimum values of #f(x) = (x + 1)/(x^2 + x + 9)# on the interval #[ 0,∞)#?
On the interval
graph{x^2+x+9=0 [10, 10, 0.4, 0.4]}
graph{x^2+x+9=0 [200, 200, 0.4, 0.4]}
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To find the absolute maximum and minimum values of f(x) = (x + 1)/(x^2 + x + 9) on the interval [0, ∞), we need to follow these steps:
 Find the critical points of f(x) by setting the derivative equal to zero and solving for x.
 Evaluate f(x) at the critical points and endpoints of the interval.
 Compare the values obtained in step 2 to determine the absolute maximum and minimum.
Let's proceed with these steps:
 Differentiating f(x) with respect to x, we get: f'(x) = [(x^2 + x + 9)(1)  (x + 1)(2x + 1)] / (x^2 + x + 9)^2 Simplifying, we have: f'(x) = (x^2 + x + 9  2x^2  3x  1) / (x^2 + x + 9)^2 f'(x) = (x^2  2x + 8) / (x^2 + x + 9)^2
Setting f'(x) equal to zero, we solve for x: x^2  2x + 8 = 0 x^2 + 2x  8 = 0 (x + 4)(x  2) = 0 x = 4 or x = 2

Now, we evaluate f(x) at the critical points and endpoints of the interval: f(0) = (0 + 1)/(0^2 + 0 + 9) = 1/9 f(2) = (2 + 1)/(2^2 + 2 + 9) = 3/15 = 1/5

Comparing the values obtained, we find: f(0) = 1/9 is the absolute minimum value on the interval [0, ∞). f(2) = 1/5 is the absolute maximum value on the interval [0, ∞).
Therefore, the absolute minimum value is 1/9 and the absolute maximum value is 1/5.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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