1) Prove that for every x>=0 and 0<a<1
x^a-ax<=1-a
2)prove that for 0<=x<=1:
x^m*(1-x)^n<= (m^m*n^n)/((m+n)^(m+n))
can someone please guide/help me in about how to solve and or approch these questions?
(I know I have already posted it before but i think it is buried beneath other questions therfore i post it again)
To solve these questions, we will use mathematical proof techniques. I will guide you step-by-step on how to approach each question.
1) Prove that for every x>=0 and 0<a<1: x^a - ax <= 1-a
Step 1: Start by stating the given inequality to be proved:
x^a - ax <= 1 - a
Step 2: Determine the key properties or theorems that may help in proving the inequality.
In this case, we can use the fact that for any positive number x, the inequality x <= 1 holds.
Step 3: Begin the proof by assuming the given inequality is true.
Start with the left-hand side of the inequality:
x^a - ax <= 1 - a
Step 4: Simplify the left-hand side expression.
Using the properties of exponents, we can rewrite x^a as e^(a * ln(x)), where ln(x) is the natural logarithm of x. The inequality becomes:
e^(a * ln(x)) - ax <= 1 - a
Step 5: Apply the property mentioned in Step 2 to simplify further.
Since x >= 0, we have ln(x) <= 0, and e^(a * ln(x)) <= 1. Therefore, we can replace e^(a * ln(x)) with a number less than or equal to 1. The inequality becomes:
1 - ax <= 1 - a
Step 6: Cancel out the common terms on both sides of the inequality.
Subtracting 1 from both sides gives:
-ax <= -a
Step 7: Divide through by -a.
Remember that a is a positive number between 0 and 1, so dividing by -a does not change the inequality sign. The inequality becomes:
x >= 1
Step 8: Confirm that the assumption made in the beginning is valid.
We started with the assumption that the given inequality is true, so we must now check that x >= 1 satisfies the original inequality: x^a - ax <= 1 - a.
Since x >= 1, the left-hand side of the inequality becomes x^a - ax >= 1 - a, which is the same as 1 - ax <= 1 - a.
Thus, the assumption is valid, and the inequality x^a - ax <= 1 - a holds for every x >= 0 and 0 < a < 1.
Now let's move on to the second question:
2) Prove that for 0 <= x <= 1: x^m*(1-x)^n <= (m^m*n^n)/((m+n)^(m+n))
Step 1: Start by stating the given inequality to be proved:
x^m * (1 - x)^n <= (m^m * n^n) / ((m + n)^(m + n))
Step 2: Determine the key properties or theorems that may help in proving the inequality.
In this case, we can use the fact that for any numbers a and b, (a+b)^k >= a^k + b^k holds for k >= 1.
Step 3: Begin the proof by assuming the given inequality is true.
Start with the left-hand side of the inequality:
x^m * (1 - x)^n <= (m^m * n^n) / ((m + n)^(m + n))
Step 4: Simplify the right-hand side expression.
Use the property mentioned in Step 2. We have:
(m^m * n^n) / ((m + n)^(m + n)) = (m^m / (m + n)^m) * (n^n / (m + n)^n)
Step 5: Apply the property mentioned in Step 2 to simplify further.
Using the property, we can rewrite the right-hand side expression as:
(m^m / ((m + n) / m)^m) * (n^n / ((m + n) / n)^n)
Step 6: Cancel out the common terms on both sides of the inequality.
Divide both sides of the original inequality by (m + n)^(m + n):
(x^m * (1 - x)^n) / ((m + n)^(m + n)) <= (m^m / ((m + n) / m)^m) * (n^n / ((m + n) / n)^n)
Step 7: Notice that ((m + n) / m)^m = (1 + n/m)^m and ((m + n) / n)^n = (1 + m/n)^n.
Therefore, the inequality becomes:
(x^m * (1 - x)^n) / ((m + n)^(m + n)) <= (m^m * n^n) / ((m + n)^(m + n))
Step 8: Since the expression on the left-hand side of the inequality matches the expression on the right-hand side, the original inequality holds.
Thus, we have proved that for 0 <= x <= 1: x^m * (1 - x)^n <= (m^m * n^n) / ((m + n)^(m + n)).
I hope this explanation helped you understand the approach to these problems. If you have any further questions, feel free to ask!