Answer:
Minimum
Need to know:
Derivative power rule: d/dx xⁿ = nxⁿ⁻¹
Derivative constant rule: d/dx c = 0
Explanation:
A way to determine whether the quadratic equation has a maximum or a minimum is by graphing it.
If we look at the image below, we notice that the graph does not go to the bottom infinitely, but it keeps going up infinitely. This means there is a minimum to this graph.
Another way to determine this is through calculus.
First, we have to find the point of extremum
Find the derivative of the given equation
d/dx 4x² - 3x - 43
You can separate the terms and solve them individually
d/dx 4x² = 4(2)(x²⁻¹) = 8x
d/dx -3x = -3(1)(x¹⁻¹) = -3(1)(1) = -3
d/dx -43 = 0
y' = 8x - 3
Set y' to equal 0
8x - 3 = 0
Add 3 to both sides
8x - 3 = 0
+ 3 + 3
8x = 3
Divide both sides by 8
8x/8 = 3/8
x = 3/8
There is only one point of extremum
Now we have to test whether the equation changes from negative to positive or positive to negative
If it changes from negative to positive, that point is a minimum
If it changes from positive to negative, that point is a maximum
To find this, pick a number less than 3/8 and plug it in the place of x in the derivative equation. We'll use 0.
y' = 8(0) - 3 = -3
Now we will do the same for a number larger than 3/8. We'll use 1
y' = 8(1) - 3 = 5
Since the lesser side of 3/8 is negative and the larger side of 3/8 is positive, that means it changes from negative to positive. This point is a minimum.