If `f:[-1,2]->B` is given by `f(x)=3x^2` then find `B` so that `f` is onto.

Answer

`ararrq,r;brarrr,s;crarrp,r;drarrr,s`

Explanation

(a) `f(x)=(x-1)^(3)(x+2)^(5)`
or `f(X)=3(x-1)^(2)(x+2)^(5)+5(x-1)^(3)(x+2)^(4)`
`=(x-1)^(2)(x+2)^(4)[3(x+2)+5(x-1)]`
`=(x-1)^(2)(x+2)^(4)[8x+1]`
sign of derivative does not changes at x=1 and x=-2
Sign of derivative hcnges at `x=-1//8` form -ve to +ve
Hence function has point of minim a
Also f(x)=0 for x=1 and x=-2
Hence function has tow points of inflection
b f(X) `=3cos x -4sin x -5le 0`
or Hence f(X) is decresing function
Also f(X)=-3sinx-4cosx =0 for infinite values of x
Hence function has infinite points inflection
(c )
From the graph x=1 is point of maxima as well as point of inflection
(d) `f(X) =(x-1)^(3//5) or f(X)=(3/5)(x-1)^(-2)/(5) ge 0`
For all real x
Also `f(X) =-(3)/(5)(2)/(5)(x-1)^(-7)/(5)` which changes sign at x=1
Hence x =1 is point of inflection