Question

Find all the critical numbers of the function\[f(x) = {x^{\dfrac{4}{5}}}{(x - 4)^2}\]?

Answer 1

Hint: To find the critical point we have to find the values which satisfy the equation, for obtaining these values we have to equate the equation to zero and then find factors these factors are the critical points of the equation. The factors obtained will always satisfy the equation if it is not satisfying, then there is some error in calculation.

Complete step by step answer:

Given equation is \[f(x) = {x^{\dfrac{4}{5}}}{(x - 4)^2}\]

Critical points are the points that satisfy the given equation, in order to obtain our critical point or roots of the equation we are equating the equation with \[0\] 

\[  {x^{\dfrac{4}{5}}} = 0,{(x - 4)^2} = 0 \]

\[ x = 0,\,(x + 4)(x + 4) = 0 \]

\[ x = 0,\,x =  - 4,\,x =  - 4 \]

The critical points of the equation is \[0,\, - 4,\, - 4\]  

Formulae Used: Equate the equation with zero so as to obtain the roots.

Additional Information: Any equation satisfies its roots i.e. when you put the root in the equation then the L.H.S of the equation equals R.H.S.

Note:

This method of finding the root is an easy method, but sometimes factorization of the equation is not possible and you are unable to draw the roots, basically that equation a graph of any particular shape like an ellipse, hyperbola, circle, etc. In dealing with such equations you have to identify that given equation belongs to which shape and then you can use graphical technique i.e. after drawing the graph every turning point in the graph shows the critical points.

For example: If you are asked to find the critical points in the equation \[f(x) = \sin x,\,[\text{no domain defined}]\] then number of critical point in this equation is infinite.