The quadratic inequality is represented by this description is y > 9x²/8 + 45x/4 + 369/8
Since the function is a parabola, we represent is as y = f(x) = ax² + bx + c
Now, given that the zeros of a parabola are −2 and −8, we have that
f(-2) = 0
a(-2)² + b(-2) + c = 0
4a - 2b + c = 0
4a - 2b = -c (1)
Also, f(-8) = 0
a(-8)² + b(-8) + c = 0
64a - 8b + c = 0
64a - 8b = -c (2)
Also, dy/dx = d(ax² + bx + c)/dx = 2ax + b
At maximum value, dy/dx = 0.
So, 2ax + b = 0
2ax = -b
x = -b/2a
Substituting this into y, we have
f(-b/2a) = a(-b/2a)² + b(-b/2a) + c
f(-b/2a) = ab²/4a² - b²/2a + c
f(-b/2a) = b²/4a - b²/2a + c
f(-b/2a) = - b²/4a + c
Since the maximum value of y is 18. So,
f(-b/2a) = - b²/4a + c = 18
- b²/4a + c = 18
c = 18 + b²/4a
Substituting c into equation (1) and (2), we have
So, in equation (1)
4a - 2b = -c (1) × 16
4a - 2b = -(18 + b²/4a) (1)
4a - 2b = -18 - b²/4a
16a² - 8ab = -72a (4)
Also, in equation (2)
64a - 8b = -c (2) × 1
64a - 8b = -(18 + b²/4a) (2)
64a - 8b = -18 - b²/4a)
256a² - 32ab = -72a (5)
Subtracting equations (4) and (5),we have
16a² - 8ab = -72a (4)
-
256a² - 32ab = -72a (5)
-240a² + 24ab = 0
-24a(10a - b) = 0
⇒ 24a = 0 or 10a - b = 0
⇒ a = 0 or 10a = b
⇒ a = 0 or b = 10a
Substituting b = 10a into equation (4), we have
16a² - 8ab = -72a (4)
16a² - 8a(10a) = -72a (4)
16a² - 80a² = -72a (4)
-64a² = -72a
-64a² + 72a = 0
-8a(8a - 9) = 0
⇒ -8a = 0 or 8a - 9 = 0
⇒ a = 0 or 8a = 9
⇒ a = 0 or a = 9/8
Substituting a into b, we have
b = 10a
b = 10 × 9/8
b = 5 × 9/4
b = 45/4
Substituting a nand b into c, we have
c = 18 + b²/4a
c = 18 + (10a)²/4a
c = 18 + 100a²/4a
c = 18 + 25a
c = 18 + 25 × 9/8
c = 18 + 225/8
c = (144 + 225)/8
c = 369/8
So, susbtituting the values of a, b and c into y, we have
y = ax² + bx + c
y = 9x²/8 + 45x/4 + 369/8
Since the equation of the parabola is y = 9x²/8 + 45x/4 + 369/8 and the region shaded is the inside of the parabola, we use the inequality sign > since the region is greater than y.
Also, since there is a solid line bounding the region, the line is not included in the inequality. so, the greater than sign > is used.
So, the shaded region is y > 9x²/8 + 45x/4 + 369/8
So, the quadratic inequality is represented by this description is y > 9x²/8 + 45x/4 + 369/8
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