need to classify to stationary points as local minimum and local maximum of g(x)
got this far
2x-7=0
4x+1=0
so x = 7/2
and x = 1/4
but bit confused on using xL and xR for each to get the local min and max..?
To determine whether each stationary point you found is a local minimum or maximum of the function g(x), you can use the values of xL and xR. xL represents a value to the left of the stationary point, and xR represents a value to the right of the stationary point.
Here's how you can determine if each point is a local minimum or maximum:
1. Start with the first stationary point, x = 7/2:
- Choose a value xL that is slightly to the left of 7/2, such as xL = 3. This means xL is smaller than 7/2.
- Choose a value xR that is slightly to the right of 7/2, such as xR = 4. This means xR is larger than 7/2.
- Substitute xL and xR into g(x) and compare the results. If g(xL) < g(x) and g(xR) < g(x), then the stationary point is a local maximum. If g(xL) > g(x) and g(xR) > g(x), then the stationary point is a local minimum. In other words, check whether the function values on both sides are greater or smaller than the function value at the stationary point.
2. Proceed with the second stationary point, x = 1/4:
- Choose a value xL that is slightly to the left of 1/4, such as xL = 0. This means xL is smaller than 1/4.
- Choose a value xR that is slightly to the right of 1/4, such as xR = 0.5. This means xR is larger than 1/4.
- Substitute xL and xR into g(x) and compare the results. If g(xL) < g(x) and g(xR) < g(x), then the stationary point is a local maximum. If g(xL) > g(x) and g(xR) > g(x), then the stationary point is a local minimum.
By comparing the function values at various points on either side of each stationary point, you can determine if it is a local minimum or maximum.