How to find local maximum and local minimum of a function?
Answers
Answer:
Find the first-order derivative of the function first, then make the derivative function equals to zero and solve for x. Once x is solved, substitute the solution of x back to the original function to obtain the y-value.
Example:
f(x)=x^2
f'(x)=2x
Let f'(x)=0
Then 2x=0, x=0
Substiture x=0 back into f(x):
f(0)=0
Therefore, local minimum occurs at x=0, y=0
The above method can be used for any function that has a stationary point (point where gradient is zero), not just quadratic function which I have used as an example.
To determine the nature of the turning point- that is, to determine whether it is local maximum or minimum, you look at the signs of the derivative (gradient) slightly towards the left and right of the turning point. Using the above example:
Let x=-1 (Slightly towards left) as well as x=1 (Slightly towards right)
SInce f'(-1)=-2 and f'(1)=2, it means that the sign of the gradient changes from negative to positive from left to right, hence it's a local minimum. Conversely, changin from positive to negative from left to right indicates a local maximum.
Hope it helps :)
Step-by-step explanation: