Find all extrema for the function
y=x^5-2x^3+3 on the interval [-10...10]
To find the extrema of a function, we need to find the points where the function reaches its maximum or minimum values.
Step 1: Find the critical points of the function.
To find the critical points, we need to find the values of x where the derivative of the function is equal to zero or undefined. Recall that critical points occur at the points where the derivative changes from positive to negative or from negative to positive.
The given function is y = x^5 - 2x^3 + 3.
Step 2: Calculate the derivative of the function.
Let's differentiate the function to find its derivative:
dy/dx = 5x^4 - 6x^2
Step 3: Find the values of x where the derivative is equal to zero or undefined.
Setting the derivative equal to zero:
5x^4 - 6x^2 = 0
Factorizing the equation:
x^2(5x^2 - 6) = 0
From this equation, we have two cases:
1. x^2 = 0
=> x = 0
2. 5x^2 - 6 = 0
=> 5x^2 = 6
=> x^2 = 6/5
=> x = ±√(6/5)
So, the critical points are x = 0, x = √(6/5), and x = -√(6/5).
Step 4: Evaluate the function at the critical points and endpoints of the given interval.
We need to plug in the critical points and the endpoints of the interval [-10,10] into the function and compare the values to find the extrema.
Evaluating at the endpoints:
For x = -10:
y = (-10)^5 - 2(-10)^3 + 3
= -10,000 + 2,000 + 3
= -8,997
For x = 10:
y = 10^5 - 2(10)^3 + 3
= 100,000 - 2,000 + 3
= 97,003
Evaluating at the critical points:
For x = 0:
y = (0)^5 - 2(0)^3 + 3
= 0 - 0 + 3
= 3
For x = √(6/5):
y = (√(6/5))^5 - 2(√(6/5))^3 + 3
≈ 9.001
For x = -√(6/5):
y = (-√(6/5))^5 - 2(-√(6/5))^3 + 3
≈ -0.001
Step 5: Analyze the values obtained.
Comparing the values obtained, we can conclude the extrema values:
The function reaches a maximum at x = 10, with a value of y ≈ 97,003.
The function reaches a minimum at x = -√(6/5), with a value of y ≈ -0.001.
The function reaches a maximum at x = √(6/5), with a value of y ≈ 9.001.
The function reaches a minimum at x = 0, with a value of y = 3.
Therefore, the extrema of the function y = x^5 - 2x^3 + 3 on the interval [-10, 10] are:
Maximum: (10, 97003)
Minimum: (-√(6/5), -0.001)
Maximum: (√(6/5), 9.001)
Minimum: (0, 3)