In this problem, we are given three inequalities that hold for a quadratic function y = ax² + bx + c. We need to find the least value of 'a' that satisfies these inequalities.


To find the least value of 'a', we need to analyze each inequality separately and find the conditions that satisfy them.


To solve this inequality, we substitute x = -1 into the quadratic function y = ax² + bx + c:

y(-1) = a(-1)² + b(-1) + c
= a + b + c

We can rewrite the inequality as:

a + b + c ≥ -4


Substituting x = 1 into the quadratic function:

y(1) = a(1)² + b(1) + c
= a + b + c

The inequality can be written as:

a + b + c ≤ 0


Substituting x = 3 into the quadratic function:

y(3) = a(3)² + b(3) + c
= 9a + 3b + c

The inequality becomes:

9a + 3b + c ≥ 5


To find the least value of 'a', we need to consider all the inequalities together.

From the first inequality, we have:

a + b + c ≥ -4

From the second inequality, we have:

a + b + c ≤ 0

From the third inequality, we have:

9a + 3b + c ≥ 5


To find the least value of 'a', we need to find the range of values that satisfy all three inequalities simultaneously.

By combining the first two inequalities, we have:

-4 ≤ a + b + c ≤ 0

By subtracting the second inequality from the first inequality, we get:

-4 - (a + b + c) ≤ a + b + c - (a + b + c) ≤ 0 - (a + b + c)

Simplifying, we get:

-4 ≤ -2(a + b + c) ≤ 0

Dividing by -2, we get:

2 ≥ a + b + c ≥ 0

From the third inequality, we have:

9a + 3b + c ≥ 5

Combining this inequality with the previous inequality, we have:

2 ≥ a + b + c ≥ 0 ≥ 5

Therefore, the least value of 'a' is 5.


The least value of 'a' that satisfies the given inequalities is 5. By analyzing each inequality separately and combining them, we found the range of values for 'a' that satisfy all three inequalities simultaneously.

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