(a)(i)f(x,y)=cos(x2+y2)
f′(x,y)x=−2xsin(x2+y2)
f′′(x,y)xx=−2sin(x2+y2)−2x.2xcos(x2+y2)=−2sin(x2+y2) −4x2cos(x2+y2)
f′(x,y)y=−2ysin(x2+y2)
f′′(x,y)yy=−2sin(x2+y2)−2y.2ycos(x2+y2)=−2sin(x2+y2) −4y2cos(x2+y2)
(ii) f(x,y)=sin(x/y)f′(x,y)x=ycos(x/y)f′′(x,y)xx=−y2sin(x/y)f′(x,y)y=−y2xcos(yx)f′′(x,y)yy=x(y2)2y2xsin(yx)y2−2ycos(yx)=y4x(xsin(yx)−2ycos(yx))
(b)f(x,y)=10–x2–y2 & ;x2+y2≤9f(x,y)=10−(x2+y2)maximum value ofx2+y2 is 9 and minimum value is 0
f(x,y) ranges between 1 to 10
(c)f(x,y)=∣x−1∣ at (0,1)
From the graph we can see that there is sharp turn at x=1 which means the function is not differentiable at x=1
(ii)y3+ysin2x+ex+y=0
In the above function no subpart of the function which is not differentiable at any point
sin , exponential and y^3 all are continuous and differentiable at all domain
Hence the above function is differentiable
d)w=xy+x,x=cost,y=sint
substituting x and y in w
w=cost∗sint+cost
w=(sin2t)/2+cost
dw/dt=cos2t−sint
puttingt=0dw/dt=1
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