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What are the minimum and maximum values of x, for which x³−x²−6x≥0? A) Minimum = 3, Maximum = 0 B) Minimum = 0, Maximum = 2 C) Minimum = 3, No maximum D) Minimum = 2, No maximum

User Vlee
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Answer:

Minimum = 3, No maximum.

Option (c) is true.

Explanation:

To find the minimum and maximum values of x for which the inequality x³−x²−6x≥0 is true, we can factor the expression and analyze its behavior.

First, let's factor the expression:

x³−x²−6x = x(x²−x−6) = x(x−3)(x+2)

The inequality x³−x²−6x≥0 can be satisfied when the expression on the left side is either greater than or equal to zero.

To determine the values of x that satisfy the inequality, we can consider the sign of each factor.

For x, the factor x is positive for x > 0, zero for x = 0, and negative for x < 0.

For (x−3), the factor (x−3) is positive for x > 3, zero for x = 3, and negative for x < 3.

For (x+2), the factor (x+2) is positive for x > -2, zero for x = -2, and negative for x < -2.

Now, let's analyze the different regions of x and determine when the inequality is satisfied:

For x < -2: In this range, all three factors are negative.

Multiplying three negative numbers results in a negative value. Therefore, the inequality is not satisfied.

For -2 < x < 0: In this range, the factor x is negative, the factor (x−3) is negative, and the factor (x+2) is positive.

Multiplying two negative numbers and one positive number results in a positive value.

Therefore, the inequality is satisfied.

For 0 < x < 3:

In this range, the factor x is positive, the factor (x−3) is negative, and the factor (x+2) is positive.

Multiplying one positive number, one negative number, and one positive number results in a negative value.

Therefore, the inequality is not satisfied.

For x > 3:

In this range, all three factors are positive.

Multiplying three positive numbers results in a positive value.

Therefore, the inequality is satisfied.

Based on the analysis above, the minimum and maximum values of x for which the inequality x³−x²−6x≥0 is true are:

Minimum = -2

Maximum = 3

Therefore,

Option (c) is true.

User Rahul Rathore
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