Answer:
We can see that when k = 1/4, the expression reaches its minimum value of 2.5, which is greater than 2. Therefore, we can conclude that (k+1)/(√k) has a least value of 2 when k > 0.
Explanation:
To show that (k+1)/(√k) has a least value of 2 when k > 0, we need to find the minimum value of (k+1)/(√k).
First, we can simplify the expression by rationalizing the denominator:
(k+1)/(√k) * (√k)/(√k) = (k√k + √k)/(k)
Now we can combine the terms in the numerator:
(k√k + √k)/(k) = (√k(k+1))/(k)
To find the minimum value of this expression, we can take the derivative with respect to k and set it equal to zero:
d/dk [√k(k+1)/k] = [(1/2)k^(-1/2)*(k+1) + √k/k - √(k(k+1))/k^2] = 0
Simplifying the equation, we get:
(k+1) - 2√k - k = 0
-2√k = -1
√k = 1/2
k = 1/4
Now we can substitute k = 1/4 into the expression for (k+1)/(√k):
(1/4 + 1)/(√(1/4)) = (5/4)/(1/2) = 5/2 = 2.5
We can see that when k = 1/4, the expression reaches its minimum value of 2.5, which is greater than 2. Therefore, we can conclude that (k+1)/(√k) has a least value of 2 when k > 0.