`\color{green} ✍️` Let `f` be a real valued function and let `c` be an interior point in the domain of `f.`
(a) `c` is called a point of local maxima if there is an `h > 0` such that
`color{red}{f (c) ≥ f (x)}`, for all x in `(c – h, c + h)`
`=>` The value f (c) is called the local maximum value of f.
(b) `=>c` is called a point of local minima if there is an `h > 0` such that
`color{red}{f (c) ≤ f (x),}` for all `x` in `(c – h, c + h)`
`=>` The value f (c) is called the local minimum value of `f .`
`color{blue}{"Note :"}` The function `f` is increasing (i.e., `(f ′(x) > 0)` in the interval `(c – h, c)` and decreasing (i.e., `(f ′(x) < 0)` in the interval `(c, c + h).`
`=>`Similarly, if `c` is a point of local minima of `f` , then the graph of `f` around `c` will be as shown in fig.
Here f is decreasing (i.e., `f ′(x) < 0)` in the interval (c – h, c) and increasing (i.e., `(f ′(x) > 0)` in the interval `(c, c + h).`
`\color{green} ✍️` Let `f` be a real valued function and let `c` be an interior point in the domain of `f.`
(a) `c` is called a point of local maxima if there is an `h > 0` such that
`color{red}{f (c) ≥ f (x)}`, for all x in `(c – h, c + h)`
`=>` The value f (c) is called the local maximum value of f.
(b) `=>c` is called a point of local minima if there is an `h > 0` such that
`color{red}{f (c) ≤ f (x),}` for all `x` in `(c – h, c + h)`
`=>` The value f (c) is called the local minimum value of `f .`
`color{blue}{"Note :"}` The function `f` is increasing (i.e., `(f ′(x) > 0)` in the interval `(c – h, c)` and decreasing (i.e., `(f ′(x) < 0)` in the interval `(c, c + h).`
`=>`Similarly, if `c` is a point of local minima of `f` , then the graph of `f` around `c` will be as shown in fig.
Here f is decreasing (i.e., `f ′(x) < 0)` in the interval (c – h, c) and increasing (i.e., `(f ′(x) > 0)` in the interval `(c, c + h).`