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.