Let f(x) =`|{:(x^(3), sinx,cosx),(6,-1,0),(p,p^(2),p^(3)):}|` where p is a constant. Then `d^(3)/dx^(3){f(x)}` at x=0 is

Let f(x)={{:(xe^(ax)",", x le0),(x+ax^2-x^3",",x gt 0):} where a is postive constant . Find the interval in which f'(X) is increasing.

Let f(x) =|{:(secx,x^(2),x),(2sinx,x^(3),2x^(2)),(tan3x,x^(2),x):}|lim_(x to 0)f(x)/(x^(4)) is equal to

Let f(x)= |{:(cosx,sinx,cosx),(cos2x,sin2x,2cos2x),(cos3x,sin3x,3cos3x):}| then find the value of f'((pi)/(2)) .

If f(x) =|{:(sinx,cosx,sinx),(cosx,-sinx,cosx),(x,1,1):}| find the value of 2[f'(0)]+[f'(1)]^(2)

Let f(x)={:{(x^(2)+4x",",-3lexle0),(-sinx",",0ltxle(pi)/(2)),(-cosx-1",",(pi)/(2)ltxlepi):} then

Let f(x) = {:{((sinx)/x + cosx, when x ne 0),(2, when x = 0):} show that f (x) is continouous at x = 0.

Let f(x) = {{:((a(1-x sin x)+b cos x + 5)/(x^(2))",",x lt 0),(3",",x = 0),([1 + ((cx + dx^(3))/(x^(2)))]^(1//x)",",x gt 0):} If f is continuous at x = 0, then (a + b + c + d) is

Let f(x) = x^3/3-x^2/2+x-16 . Find f’(0), f’(-1).

A random variable X takes the values 0,1,2,3,..., with prbability P(X=x)=k(x+1)((1)/(5))^x , where k is a constant, then P(X=0) is.

Let f(x)=(x^(2)-2x+1)/(x+3),f i n d x: (i) f(x) gt 0 (ii) f(x) lt 0