The polynomial function is defined by f(x)=4x^4+2x^3-8x^-5x+2.
Use a calculator to find all the points (x,f(x)) where there is a local maximum.
Round to the nearest hundredth.
f '(x) = 16x^3 + 6x^2 - 16x - 5 , ( I assumed the second last term was -8x^2 )
16x^3 + 6x^2 - 16x - 5 = 0
zeros for this equation exist at
appr. -1,1 and -.3
f ''(x) = 48x^2+ 12x - 16
both f ''(1) and f ''(-1) > 0
but f ''(-.3) < 0
so when x = -.3 (appr), we have a local maximum
I will let you find f(-.3)
To find the points where there is a local maximum for the given polynomial function f(x), we can follow these steps:
1. First, we need to find the derivative of the function f(x). The derivative will give us information about the slope of the function at different points.
2. Once we have the derivative, we need to set it equal to zero and solve for x. This will give us the x-values of the local maximum points.
3. Finally, we substitute the x-values we found in step 2 back into the original function f(x) to find the corresponding y-values (f(x)).
Let's go through these steps:
Step 1: Find the derivative of f(x)
The derivative of f(x) can be found by taking the derivative of each term separately.
Given: f(x) = 4x^4 + 2x^3 - 8x^-5x + 2
To find the derivative, we differentiate each term:
f'(x) = (4 * 4x^3) + (3 * 2x^2) - (8 * -5x^(-6)) + (0) + (0)
Simplifying, we have:
f'(x) = 16x^3 + 6x^2 + 40x^(-6)
Step 2: Set the derivative equal to zero and solve for x
To find the x-values where the derivative is equal to zero, we set f'(x) = 0 and solve for x.
16x^3 + 6x^2 + 40x^(-6) = 0
Unfortunately, the equation involving both positive and negative exponents (40x^(-6)) introduces complex solutions. Since we want to round to the nearest hundredth, we focus on real solutions. However, in this case, there are no real solutions for x.
Step 3: Substitute x-values back into the original function
Since we couldn't find any real solutions for x, there are no local maximum points for the given polynomial function, f(x) = 4x^4 + 2x^3 - 8x^-5x + 2.
Therefore, there are no points (x, f(x)) where there is a local maximum.