Find all the intervals on which the function is concave upward: f(x)= 4x^2-x/x^4
Is (4x^2 -x) the numerator?
If so, you should have used parentheses around it.
A function is concave upward where the second derivative is positive.
To determine the intervals on which the function f(x) = (4x^2 - x) / x^4 is concave upward, we need to consider the second derivative of the function.
First, let's find the first and second derivatives of f(x).
The first derivative, f'(x), can be found using the quotient rule:
f'(x) = [(x^4) * (d/dx)(4x^2 - x) - (4x^2 - x) * (d/dx)(x^4)] / (x^4)^2
Simplifying the above expression, we get:
f'(x) = [(4x^6 - x^5) - (12x^3 - 4x^2)] / x^8
f'(x) = (4x^6 - x^5 - 12x^3 + 4x^2) / x^8
Next, let's find the second derivative, f''(x), by differentiating f'(x):
f''(x) = (d/dx)[(4x^6 - x^5 - 12x^3 + 4x^2) / x^8]
To simplify the expression, we can rewrite the function as:
f(x) = 4x^(-2) - x^(-3) - 12x^(-5) + 4x^(-6)
Now, let's find f''(x) using the power rule:
f''(x) = [d/dx (4x^(-2)) - d/dx (x^(-3)) - d/dx (12x^(-5)) + d/dx (4x^(-6))]
f''(x) = -8x^(-3) + 3x^(-4) + 60x^(-6) - 24x^(-7)
Simplifying the expression, we get:
f''(x) = -8/x^3 + 3/x^4 + 60/x^6 - 24/x^7
Now that we have the second derivative, we can determine the intervals on which the function is concave upward by finding where f''(x) > 0.
Finding the common denominator, we get:
f''(x) = (-8x^4 + 3x^3 + 60 - 24x) / (x^7)
Next, we look for the values of x for which f''(x) > 0. We set the numerator equal to zero and solve for x:
-8x^4 + 3x^3 + 60 - 24x > 0
Factorizing the expression, we get:
(x - 3)(8x^3 + 24x^2 + 8x - 20) > 0
To find the intervals on which f''(x) > 0, we can use a sign chart or synthetic division to test different intervals. Let's test intervals (-∞, -1), (-1, 0), (0, 3), and (3, +∞).
For x ∈ (-∞, -1):
Let's substitute x = -2 into the expression 8x^3 + 24x^2 + 8x - 20:
8(-2)^3 + 24(-2)^2 + 8(-2) - 20 = -8 + 96 - 16 - 20 = 52
Since the expression evaluates to a positive value, f''(x) > 0 for x ∈ (-∞, -1).
For x ∈ (-1, 0):
Let's substitute x = -0.5 into the expression 8x^3 + 24x^2 + 8x - 20:
8(-0.5)^3 + 24(-0.5)^2 + 8(-0.5) - 20 = -0.5 + 3 - 4 - 20 = -21.5
Since the expression evaluates to a negative value, f''(x) < 0 for x ∈ (-1, 0).
For x ∈ (0, 3):
Let's substitute x = 1 into the expression 8x^3 + 24x^2 + 8x - 20:
8(1)^3 + 24(1)^2 + 8(1) - 20 = 8 + 24 + 8 - 20 = 20
Since the expression evaluates to a positive value, f''(x) > 0 for x ∈ (0, 3).
For x ∈ (3, +∞):
Let's substitute x = 4 into the expression 8x^3 + 24x^2 + 8x - 20:
8(4)^3 + 24(4)^2 + 8(4) - 20 = 512 + 384 + 32 - 20 = 908
Since the expression evaluates to a positive value, f''(x) > 0 for x ∈ (3, +∞).
Therefore, the intervals on which the function f(x) = (4x^2 - x) / x^4 is concave upward are (-∞, -1) and (3, +∞).
To determine the intervals on which the function is concave upward, we need to analyze the second derivative of the function.
First, let's find the first derivative of the function f(x).
Given function: f(x) = (4x^2 - x) / x^4
To find the first derivative, we will apply the quotient rule:
Let u = 4x^2 - x and v = x^4
Using the quotient rule formula:
f'(x) = (v * u' - u * v') / v^2
Where u' and v' are the derivatives of u and v respectively.
Differentiating u:
u' = 8x - 1
Differentiating v:
v' = 4x^3
Plugging these values into the quotient rule formula:
f'(x) = (x^4 * (8x - 1) - (4x^3) * (4x^2 - x)) / (x^4)^2
Simplifying the equation:
f'(x) = (8x^5 - x^4 - 16x^5 + 4x^4) / x^8
f'(x) = -8x^5 + 3x^4 / x^8
Now, we need to find the second derivative by differentiating the first derivative with respect to x.
Differentiating f'(x) = -8x^5 + 3x^4 / x^8:
f''(x) = (d/dx)(-8x^5 + 3x^4 / x^8)
Using the quotient rule again:
f''(x) = (x^8 * (-40x^4 + 12x^3) - (-8x^5 + 3x^4) * 8x^7) / (x^8)^2
Simplifying the equation:
f''(x) = (-40x^12 + 12x^11 + 64x^12 - 24x^11) / x^16
f''(x) = 24x^12 - 12x^11 / x^16
To find the intervals on which the function is concave upward, we need to determine where the second derivative is positive (greater than 0).
Setting f''(x) > 0:
24x^12 - 12x^11 > 0
Dividing both sides by 12x^11 (which is always positive):
2x - 1 > 0
Simplifying the equation:
2x > 1
x > 1/2
Thus, the function f(x) = (4x^2 - x) / x^4 is concave upward for all values of x greater than 1/2.