1.

Let f:Rto R be given by fx={{:(x^5+5x^4+10x^3,+10^2+3x+1,,xlt0,,),(,x^2-x+1,,0lexlt1,,),((2)/(3)x^3-4x^2+7x-(8)/(3),,,1lexlt3,,),((x-2)log_e(x-2),-x+(10)/(3),,xge3,,):} Then which of the following options is/are correct ?

Answer»

f is increasing on `(-oo,0)`
f' is NOT differentiable at `x=1`
f' has a local maximum at `x=1`

Solution :GIVEN function `f:R to R` is
`f(x)=[{:(x^25x^4+10x^3+10x^2+3x+1",",xlt 0),(""x^2-x-x+1",",0lexlt1),(""(2)/(3)x^3-4x^2+7x-(8)/(3)",",1lexlt3),(" "(x-2)log_e(x-2)-x+(10)/(3)",",xge3):}`
So,`f(x)=[{:(5x^5+20x^3+30x^2+20x+3",",xlt 0),(""2x-1",",0ltxlt1),(""2x^2-8x+7", ",1ltxlt3),(""log_e(x-2)",",xgt3):}`
At `x=1,f^(n)(1^-)=2 gt0 and f^n(1^+)=4-8=-4lt0`
` therefore f'(x)` is not differentiable at `x=1`, and
f'(x) has a local maximum at `x=1`.
For `x in (- oo ,0)`
`f'(x)=5x^4+20x^3+30x^2+20x+3`
and since, `f'(-1)=5-20+30-20+3=-2lt 0`
So, `f(x)` is not increasing on `x in (- oo, 0)`.
Now, as the range of function `f(x)` is R, so f is onto function.
Hence, options (B) , (c) and (d) are correct.


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