maxima and minima class 12, Example Formula 12th Class, function, Problem
* Maxima and Minima:
Maxima of a function f(x):
A Function `f(x)` is said to have a maxima at `x=c` if the value of the function at `x=c` is greater than any other value of `f(x)` in a `\delta`-neighborhood of c. That is for a small `\delta>0` and for
`x \in (c-\delta, c+\delta)` we have `f(c) > f(x)`. The value of `f(c)` is called a maxima of `f(x)`. Thus the function f(x) have maxima at `x=c` if `f(x)` is increasing in
`c-\delta < x < c+\delta` and decreasing in `c<x < c+\delta`. Applications of Derivatives Maxima and Minima Mathematics
Minima of a function of f(x):
A function f(x) is said to have minima at x=c if the value of the function at x=c is less than any other value of f(x) in a`\delta`-neighborhood of c.That is for a small `\delta>0` and for `x \in (c-\delta,c+\delta)` we have `f(c) < f(x)`. The value of f(c) is called a minima of `f(x)` .Thus the function `f(x)` will have minima at `x=c` if `f(x)` is decreasing in `c-\delta < x < c` and increasing in `c < x < c+\delta`.
If `f^{'}(c)=0` then at x=0 the function is neither increasing nor decreasing, such a point on the curve is called turning point or stationary point of the function. Any point at which the tangent to the graph is horizontal is a turning point. We can locate the turn points by looking for points at which
`\frac{dy}{dx}=0`.
At these points if the function has Maxima and Minima then these are called extreme values of the function.
Applications of Derivatives Maxima and Minima Mathematics
NOTE:
A maxima `f(x)` has maxima at `x=c` if
(i) `f^{'}(c)=0`
(ii) `f^{'}(c-h) > 0 ` [`f(x)` is an increasing for values of `x < c`]
(iii) `f^{'}(c+h) < 0` [`f(x)` is decreasing for values of `x > c`]
where h is a small positive number.
A function `f(x)` has a minima at x=c if
(i) `f^{'}(c)=0`
(ii) `f^{'}(c-h) < 0` [`f(x)` is decreasing for values of `x < 0`]
(iii)`f^{'}(c+h) > 0` [`f(x)` is increasing for values of `x > 0`]
where h is a small positive number.
Note-
If `f^{'}(c) = 0` and `f^{'}(c-h) > 0` , `f^{'}(c+h) > 0` or `f^{'}(c-h) < 0` , `f^{'}(c+h) < 0` then `f(c)` is neither maxima nor minima. In such a case `x = c` is called a point of inflexion,
e.g., `f(x)=x^3` , `f(x) = x^5 ` in [ -2 , 2 ]
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| Applications of Derivatives Maxima and Minima |
 12th Class){kind=link}