Consider the function y = 3x5 – 25x3 + 60x + 1. Use the first derivative test to decide whether this function has a maximum at x = 1
y' = 15x^4 - 75x^2 + 60
So, is y'(1) = 0?
To use the first derivative test to determine whether the function has a maximum at x = 1, we need to follow these steps:
Step 1: Find the first derivative of the function.
Step 2: Find the critical points by setting the first derivative equal to zero.
Step 3: Determine the sign of the first derivative on each interval between the critical points.
Step 4: Use the signs of the first derivative to determine whether there is a maximum or minimum at x = 1.
Let's proceed with these steps one by one.
Step 1: Find the first derivative of the function.
To find the first derivative, we differentiate the function with respect to x. For the given function y = 3x^5 – 25x^3 + 60x + 1, the derivative is:
dy/dx = 15x^4 - 75x^2 + 60
Step 2: Find the critical points by setting the first derivative equal to zero.
To find the critical points, we set the first derivative equal to zero and solve for x:
15x^4 - 75x^2 + 60 = 0
Step 3: Determine the sign of the first derivative on each interval between the critical points.
To determine the sign of the first derivative, we evaluate the first derivative at a test point in each interval.
For x < -1, we can choose x = -2 as a test point:
dy/dx = 15(-2)^4 - 75(-2)^2 + 60 = 60 > 0
For -1 < x < 1, we can choose x = 0 as a test point:
dy/dx = 15(0)^4 - 75(0)^2 + 60 = 60 > 0
For x > 1, we can choose x = 2 as a test point:
dy/dx = 15(2)^4 - 75(2)^2 + 60 = -180 < 0
Step 4: Use the signs of the first derivative to determine whether there is a maximum or minimum at x = 1.
Since the sign of the first derivative changes from positive to negative as we move from left to right of x = 1, this means that the function has a maximum at x = 1.
In conclusion, the function y = 3x^5 – 25x^3 + 60x + 1 has a maximum at x = 1.
To use the first derivative test, we need to find the first derivative of the function and analyze the sign of the derivative around the critical point.
Let's find the first derivative of the function y = 3x^5 - 25x^3 + 60x + 1.
Step 1: Find the derivative
To find the derivative of y, we differentiate each term of the function separately.
The derivative of the term 3x^5 is: d(3x^5)/dx = 15x^4
The derivative of the term -25x^3 is: d(-25x^3)/dx = -75x^2
The derivative of the term 60x is: d(60x)/dx = 60
The derivative of the constant term 1 is: d(1)/dx = 0
Therefore, the derivative of y is: y' = 15x^4 - 75x^2 + 60.
Step 2: Analyze the sign of the derivative around x = 1
To determine whether the function has a maximum or minimum at x = 1, we need to find the sign of the derivative on each side of x = 1.
Substitute x = 1 into the derivative, y':
y'(1) = 15(1)^4 - 75(1)^2 + 60 = 15 - 75 + 60 = 0.
Since y'(1) is equal to 0, this does not provide conclusive information about the slope and concavity of the function.
To further analyze the sign of the derivative, we can look at the intervals on either side of x = 1.
Choose a test point in the interval (0, 1). For example, let's select x = 0.5. Substitute this value into the derivative, y':
y'(0.5) = 15(0.5)^4 - 75(0.5)^2 + 60 = 15(0.0625) - 75(0.25) + 60
= 0.9375 - 18.75 + 60
= 42.1875.
Since y'(0.5) is positive, the derivative is positive on the interval (0, 1). This means the function is increasing on the interval (0, 1).
Now, choose a test point in the interval (1, ∞). For example, let's select x = 2. Substitute this value into the derivative, y':
y'(2) = 15(2)^4 - 75(2)^2 + 60 = 15(16) - 75(4) + 60
= 240 - 300 + 60
= 0.
Since y'(2) is equal to 0, this does not provide conclusive information about the slope and concavity of the function.
From the analysis, we can see that the function is increasing on the interval (0, 1) and the derivative is non-negative.
Therefore, based on the first derivative test, we cannot conclude whether the function has a maximum or minimum at x = 1. Further analysis, such as the second derivative test, is needed.