Question

Find sup { x∈R, x²+3<4x }. Justify the answer

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Hank

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To find the supremum of the set {x∈R, x²+3<4x}, we need to determine the upper bound of the set and then verify if it is the least upper bound.

Step 1: Let's solve the inequality x²+3 < 4x:

Subtracting 4x from both sides, we get:

x² - 4x + 3 < 0

Step 2: To solve the quadratic inequality, we need to factorize the expression x² - 4x + 3:

(x - 3)(x - 1) < 0

Step 3: We need to find the critical points where the inequality changes sign. The critical points occur when (x - 3)(x - 1) = 0. So, x = 1 or x = 3.

Step 4: We plot these critical points on a number line:

x < 1 1 < x < 3 x > 3

-|---o-------|------o---|----|---->

Step 5: Now, let's check the sign of the expression (x - 3)(x - 1) in each interval:

For x < 1: (-)(-) = +

For 1 < x < 3: (+)(-) = -

For x > 3: (+)(+) = +

Step 6: Since the inequality x² + 3 < 4x is true for x < 1 and x > 3, we need to find the maximum value of x within this range.

Step 7: From the number line, we can see that x < 1 is an open interval, which means it does not have a maximum value. However, for x > 3, the maximum value is 3.

Answer: Therefore, the supremum of the set {x∈R, x²+3<4x} is 3.

Step 1: Let's solve the inequality x²+3 < 4x:

Subtracting 4x from both sides, we get:

x² - 4x + 3 < 0

Step 2: To solve the quadratic inequality, we need to factorize the expression x² - 4x + 3:

(x - 3)(x - 1) < 0

Step 3: We need to find the critical points where the inequality changes sign. The critical points occur when (x - 3)(x - 1) = 0. So, x = 1 or x = 3.

Step 4: We plot these critical points on a number line:

x < 1 1 < x < 3 x > 3

-|---o-------|------o---|----|---->

Step 5: Now, let's check the sign of the expression (x - 3)(x - 1) in each interval:

For x < 1: (-)(-) = +

For 1 < x < 3: (+)(-) = -

For x > 3: (+)(+) = +

Step 6: Since the inequality x² + 3 < 4x is true for x < 1 and x > 3, we need to find the maximum value of x within this range.

Step 7: From the number line, we can see that x < 1 is an open interval, which means it does not have a maximum value. However, for x > 3, the maximum value is 3.

Answer: Therefore, the supremum of the set {x∈R, x²+3<4x} is 3.

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