Y=5x^4 -24x^3+24x^2+17 is concave down for which interval?

A) x<0
B) x>0
C) x<-2 or x>-2/5
D) x<2/5 or x>2
E) 2/5 < x < 2

I know you take the second derivative...but I keep getting stuck with something that won't factor right?

Never mind, I got it!

To determine the concavity of a function, we need to find the second derivative and analyze its sign. Let's find the second derivative of the given function:

Given function: y = 5x^4 - 24x^3 + 24x^2 + 17

1. Find the first derivative of the function:

y' = d/dx (5x^4 - 24x^3 + 24x^2 + 17)
= 20x^3 - 72x^2 + 48x

2. Find the second derivative by differentiating y' with respect to x:

y'' = d/dx (20x^3 - 72x^2 + 48x)
= 60x^2 - 144x + 48

Now, let's analyze the sign of the second derivative to determine the concavity:

1. Consider the discriminant of the quadratic equation 60x^2 - 144x + 48 = 0:

Discriminant = (-144)^2 - 4(60)(48) = 20736 - 11520 = 9216

The discriminant is positive (9216 > 0), meaning the quadratic equation has two distinct real roots. Therefore, the quadratic expression 60x^2 - 144x + 48 cannot be factored further.

2. Next, find the critical points by setting the second derivative equal to zero and solving for x:

60x^2 - 144x + 48 = 0

Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/2a, where a = 60, b = -144, c = 48:

x = (-(-144) ± √((-144)^2 - 4(60)(48))) / (2(60))
= (144 ± √(20736 - 11520)) / 120
= (144 ± √(9216)) / 120
= (144 ± 96) / 120

x1 = (144 + 96) / 120 = 240 / 120 = 2
x2 = (144 - 96) / 120 = 48 / 120 = 2/5

The critical points are x = 2 and x = 2/5.

3. Now, determine the intervals of concavity by testing values from each interval in the second derivative.

Choosing values within each interval:

For x < 2/5, pick x = 0.
y''(0) = 60(0^2) - 144(0) + 48 = 48 > 0
Since y''(0) > 0, the function is concave up for x < 2/5.

For 2/5 < x < 2, pick x = 1.
y''(1) = 60(1^2) - 144(1) + 48 = -36 < 0
Since y''(1) < 0, the function is concave down for 2/5 < x < 2.

For x > 2, pick x = 3.
y''(3) = 60(3^2) - 144(3) + 48 = 0
Since y''(3) = 0, we cannot determine the concavity for x > 2 based on this value.

Therefore, the function is concave down for the interval 2/5 < x < 2.

The correct answer is E) 2/5 < x < 2.

To determine the concavity of the function y = 5x^4 - 24x^3 + 24x^2 + 17, you are correct that we need to take the second derivative. Let's go through the steps to find the second derivative and then identify the concave down interval:

Step 1: Find the first derivative:
To find the first derivative, we differentiate the function y = 5x^4 - 24x^3 + 24x^2 + 17 with respect to x.

dy/dx = 20x^3 - 72x^2 + 48x

Step 2: Find the second derivative:
To find the second derivative, we differentiate the first derivative (dy/dx) with respect to x.

d^2y/dx^2 = d/dx(20x^3 - 72x^2 + 48x)
= 60x^2 - 144x + 48

Step 3: Set up the inequality to find the concave down interval:
To determine the concave down interval, we need to find where the second derivative is negative. This is because a function is concave down when its second derivative is negative.

Setting d^2y/dx^2 < 0, we have:
60x^2 - 144x + 48 < 0

Step 4: Solve the inequality:
To solve the inequality, we can use different methods such as factoring, completing the square, or using the quadratic formula. In this case, factoring might not be the easiest approach, so we'll use the quadratic formula.

Using the quadratic formula, we have:
x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation 60x^2 - 144x + 48 = 0, the values of a, b, and c are:
a = 60, b = -144, c = 48

Applying the quadratic formula, we get:
x = (144 ± √((-144)^2 - 4*(60)*(48))) / (2*(60))

Simplifying:
x = (144 ± √(20736 - 11520)) / 120
x = (144 ± √(9216)) / 120
x = (144 ± 96) / 120

This gives us two possible values for x:
x1 = (144 + 96) / 120 = 240 / 120 = 2
x2 = (144 - 96) / 120 = 48 / 120 = 0.4

Step 5: Identify the concave down interval:
Now that we have the roots of the quadratic equation (x = 2 and x = 0.4), we can use these values to determine the concave down interval.

The concave down interval is the range of x-values where the function is concave down, which is when the second derivative is negative (d^2y/dx^2 < 0).

Plugging in test values for x from each of the answer choices, we can determine the concave down interval:

A) For x < 0: Substitute x = -1 into the second derivative.
d^2y/dx^2 = 60(-1)^2 - 144(-1) + 48 = 60 + 144 + 48 = 252 > 0
Since the second derivative is positive, the interval x < 0 is not concave down.

B) For x > 0: Substitute x = 1 into the second derivative.
d^2y/dx^2 = 60(1)^2 - 144(1) + 48 = 60 - 144 + 48 = -36 < 0
This means the interval x > 0 is concave down.

C) For x < -2 or x > -2/5: Substitute x = -3 into the second derivative.
d^2y/dx^2 = 60(-3)^2 - 144(-3) + 48 = 540 + 432 + 48 = 1020 > 0
Since the second derivative is positive, the interval x < -2 or x > -2/5 is not concave down.

D) For x < 2/5 or x > 2: Substitute x = 1 into the second derivative.
d^2y/dx^2 = 60(1)^2 - 144(1) + 48 = 60 - 144 + 48 = -36 < 0
This means the interval x < 2/5 or x > 2 is concave down.

E) For 2/5 < x < 2: Substitute x = 1 into the second derivative.
d^2y/dx^2 = 60(1)^2 - 144(1) + 48 = 60 - 144 + 48 = -36 < 0
This means the interval 2/5 < x < 2 is concave down.

From the analysis, we can conclude that the function y = 5x^4 - 24x^3 + 24x^2 + 17 is concave down for intervals B) x > 0, D) x < 2/5 or x > 2, and E) 2/5 < x < 2.