Question

How do I find the local maxima and minima of a function?

Answer 1

As we know that maxima and minima are the most important part of calculus. The maxima and minima of a function also collectively known as extrema are the largest and smallest value of a function, either within a given range or on the entire domain. These are the Latin words which means the maximum and minimum value of a function.

Complete step by step solution:
Let $f(x)$be a real function of a real variable which is defined in $(a,b)$ and differentiable at the point ${x_0} \in (a,b)$. We should remember that there is a necessary condition for ${x_0}$to be a local minimum or maximum which is $f'({x_0}) = 0$.
If $f(x)$ is differentiable in the entire interval, or at least in an interval around ${x_0}$, then the condition is that ${x_0}$is a local minimum around if $f'({x_0}) = 0$and then there is any number like $\delta$ which gives:
$x \in ({x_0} - \delta ,{x_0}) \Rightarrow f'(x) \leqslant 0$ and $x \in ({x_0},{x_0} + \delta ) \Rightarrow f'(x) \geqslant 0$ .
In the above case we can see that $f(x)$ is decreasing on the left of ${x_0}$ and increasing on the right, which gives that ${x_0}$ is a relative minimum. And then ${x_0}$ is a local maximum if:
$x \in ({x_0} - \delta ,{x_0}) \Rightarrow f'(x) \geqslant 0$ and $x \in ({x_0},{x_0} + \delta ) \Rightarrow f'(x) \leqslant 0$.
In both the cases we can see that the necessary condition is that the derivative of $f(x)$ changes it’s sign around ${x_0}$.
If the function $f(x)$ also has a second derivative in an interval around ${x_0}$, this is equal to the conditions:
$f'({x_0}) = 0$ and $f''({x_0}) > 0 \Rightarrow {x_0}$ is a local minimum
And
$f'({x_0}) = 0$and $f''({x_0}) < 0 \Rightarrow {x_0}$is a local maximum.

Hence the answer is $f '({x_0}) = 0$ and $f''({x_0}) > 0 \Rightarrow {x_0}$ is a local minima and $f'({x_0}) = 0$and $f''({x_0}) < 0 \Rightarrow {x_0}$is a local maxima.

Note: We should keep in mind the process to find the local maxima and minima which is that we should first find the solutions of the equation i.e. $f'(x) = 0$ which is also called the critical points. And then solve the inequality $f'(x) \leqslant 0$ to check if it changes the sign around the critical points and then calculate $f''(x)$ and then check for its value.