For what values of x is the graph
y=2/(5-x) concave upward?
x<5
x>-5
x>5
no values of x
y" = 4/(5-x)^3
f is concave up when f" > 0
So, what do you think?
To determine the concavity of a function, we need to examine its second derivative. Let's start by finding the first and second derivatives of the given function, y = 2/(5 - x), and then analyze the sign of the second derivative to determine concavity.
Step 1: Find the first derivative (dy/dx):
To find the first derivative, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the first derivative f'(x) can be calculated as:
f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2.
In this case, g(x) = 2 and h(x) = 5 - x.
Using the quotient rule, let's calculate the first derivative:
y' = [(2)(-1) - (2)(-1)] / (5 - x)^2
= -4 / (5 - x)^2.
Step 2: Find the second derivative (d^2y/dx^2):
To find the second derivative, we differentiate the first derivative with respect to x.
d^2y/dx^2 = d/dx [-4 / (5 - x)^2]
= [(-4)(-2)(5 - x)] / (5 - x)^4
= 8 / (5 - x)^3.
Step 3: Determine the concavity:
To determine the concavity, we need to analyze the sign of the second derivative.
When the second derivative is positive, the graph is concave upward.
When the second derivative is negative, the graph is concave downward.
In this case, the second derivative is 8 / (5 - x)^3.
Since the denominator (5 - x)^3 is always positive, the sign of the second derivative is solely determined by the numerator, which is 8.
When 8 is positive, the graph is concave upward.
When 8 is negative, the graph is concave downward.
Since 8 is always positive, regardless of the value of x, the graph y = 2/(5 - x) is concave upward for all values of x.
Therefore, the correct answer is: No values of x. The graph is concave upward for all x.