Maxima and Minima in Calculus - Practice Questions & MCQ

Maxima and Minima of a Function

Let $y=f(x)$ be a real function defined at $x=a$. Then the function $f(x)$ is said to have a maximum value at $x=a$ if $f(x) \leq f(a) \quad \forall a \in R$.
And also the function $f(x)$ is said to have a minimum value at $x=a$, if $f(x) \geq f(a) \quad \forall a \in R$

Concept of Local Maxima and Local Minima 

The function $f(x)$ is said to have local maxima (or maxima) at a point 'a' if the value of $f(x)$ at ' $a$ ' is greater than its values for all $x$ in a small neighbourhood of ' $a$ '. In other words, $f(x)$ has a maxima at $x=$ ' $a$ ', if $f(a+h) \leq f(a)$ and $f(a-h) \leq f(a)$, where $h>0$ (very small quantity).

The function $f(x)$ is said to have local minima (or minima) at a point 'b' if the value of $f(x)$ at 'b' is less than its values for all $x$ in a small neighbourhood of 'b'. In other words, $f(x)$ has a maximum at $x=$ 'b', if $f(b+h) \geq f(b)$ and $f(b-h) \geq f(b)$, where $h>0$ (very small quantity).