Concept of local maximum and local minimum
Let y = f(x) be a function defined at x = a and also in the vicinity of the point x = a. Then, f(x) is said to have a local maximum at x = a, if the value of the function at x = a is greater than the value of the function at the neighboring points of x = a.
Mathematically, f(a) > f(a â€“ h) and f(a) > f(a + h), where h > 0. Similarly, f(x) is said to have a local minimum at x = a, if the value of the function at x = a is less than the value of the function at the neighboring points of x = a. Mathematically, f(a) < f(a â€“ h) and f(a) < f(a + h), where h > 0.
A local maximum or a local minimum is also called a local extremum.
Test for local maximum/minimum
We have two cases to consider:
- Test for local maximum/minimum at x = a if f(x) is differentiable at x = a: If f(x) is differentiable at x = a and if it is a critical point of the function (i.e., fâ€²(a) = 0) then we have the following three tests to decide whether f(x) has a local maximum or local minimum or neither at x = a.
- First derivative test: If fâ€²(a) = 0 and fâ€²(x) changes its sign while passing through the point x = a, then
- f(x) would have a local maximum at x = a if fâ€²(a â€“ 0) > 0 and fâ€²(a + 0) < 0. It means that fâ€²(x) should change its sign from positive the to negative.
- f(x) would have local minimum at x = a if fâ€²(a â€“ 0) < 0 and fâ€²(a + 0) > 0. It means that fâ€²(x) should change its sign from the negative and positive.
- If f(x) does not change its sign while passing through x = a, then f(x) would have neither a maximum nor a minimum at x = a.
- Second derivative test: This test is basically the mathematical representation of the first derivative test. It simply says that
- If fâ€²(a) = 0 and fâ€²â€²(a) < 0, then f(x) would have a local maximum at x = a.
- If fâ€²(a) = 0 and fâ€²â€²(a) > 0, then f(x) would have a local minimum at x = a.
- If fâ€²(a) = 0 and fâ€²â€²(a) = 0, then this test fails and the existence of a local maximum/minimum at x = a is decided on the basis of the nth derivative test.
- First derivative test: If fâ€²(a) = 0 and fâ€²(x) changes its sign while passing through the point x = a, then
- Test for local maximum/minimum at x = a if f(x) is not differentiable at x = a:
Concept of global maximum/minimum
Let y = f(x) be a given function with domain D. Let [a, b] âŠ† D. Global maximum/minimum of f(x) in [a, b] is basically the greatest/least value of f(x) in [a, b]. Global maximum and minimum in [a, b] would occur at critical point of f(x) within [a, b] or at the end points of the interval.
Global maximum/minimum in [a, b]
In order to find the global maximum and minimum of f(x) in [a, b], find out all the critical points of f(x) in (a, b). Let c_{1}, c_{2}, â€¦, c_{n}_{ }be the different critical points. Find the value of the function at these critical points. Let f(c_{1}), f(c_{2}), â€¦, f(c_{n}) be the values of the function at critical points.
Say, M_{1 }= max{f(a), f(c_{1}), f(c_{2}),â€¦, f(c_{n}), f(b)} and M_{2} = min{f(a), f(c_{1}), f(c_{2}), â€¦, f(c_{n}), f(b)}. Then M_{1} is the greatest value of f(x) in [a, b] and M_{2} is the least value of f(x) in [a, b]
Global maximum/minimum in (a, b)
Method for obtaining the greatest and least values of f(x) in (a, b) is almost same as the method used for obtaining the greatest and least values in [a, b], however with a caution.
Let y = f(x) be a function and c_{1}, c_{2}, â€¦, c_{n} be the different critical points of the function in (a, b).
Let M_{1} = max. {f(c_{1}), f(c_{2}), f(c_{3}), â€¦, f(c_{n})} and M_{2} = min.{f(c_{1}), f(c_{2}), f(c_{3}), â€¦, f(c_{n})}. Now if or < M_{2}, f(x) would have not global maximum (or global minimum ) in (a, b) .
This means that if the limiting values at the end points are greater than M_{1} or less than M_{2}, then f(x) would not have global maximum/minimum in (a, b). On the other hand if M_{1} > and M_{2} < , then M_{1} and M_{2} would respectively be the global maximum and global minimum of f(x) in (a, b).
Notes:
- For a continuous function maximum and minimum value occurs alternately.
- If a function is discontinuous at a point x = a it may have maximum value although it decreases on the left and increases on the right side of x = a.