The total number of local maxima and local minima of the function
`f(x)={:{((2+x)^(3)",",-3ltxle-1),(x^((2)/(3))",",-1ltxlt2):}is`

Options

A. 0
B. 1
C. 2
D. 3

Answer

C

Explanation

Given that
`f(x)={{:((2+x)^3"," -3ltxle-1),(x^(2//3)",      " -1ltxlt2):}`
`rArr f'(x) = {{:(3(2+x)^2","-3ltxlt-1),(2/3x^(-1//3",      "-1ltxlt2)):}`
Clearly , f'(x) changes its sin from positive to negative as x passes through x=-1 from left to right
So, f(x) attains a local maximum at x=-1
We observe that `f'(x)gt0 " for all " x in (-3,-1) and f'(x) lt0 " for " x in (-1,0) ,brgt Also , `f'(x) gt0 " for "x in (0,2) ` But f'(0)does not exit
So , f(x) atttains a local minimum at x=0
Hence the total number of a local maxima and local minima is 2