18/e^{4}
27/e^{2}
9/e^{2}
3 log 3-2
B.
27/e^{2}
D.
does not exist
equal
equal
equal
A.
does not exist
A function y = f(x) has a second order derivative f″(x) = 6(x – 1). If its graph passes through the point (2, 1) and at that point the tangent to the graph is y = 3x – 5, then the function is
(x-1)^{2}
(x-1)^{3}
(x+1)^{3}
(x+1)^{2}
B.
(x-1)^{3}
f''(x) = 6(x - 1) f'(x) = 3(x - 1)2 + c ........ (i)
At the point (2, 1) the tangent to graph is y = 3x - 5 Slope of tangent = 3
∴ f'(2) = 3(2 - 1)2 + c = 3
3 + c = 3
⇒ c = 0
∴ From equation (i) f'(x) = 3(x - 1)^{2}
f'(x) = 3(x - 1)2 f(x) = (x - 1)3 + k ...... (ii)
Since graph passes through (2, 1)
∴ 1 = (2 - 1)^{2} + k k = 0
∴ Equation of function is f(x) = (x - 1)^{3}
A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness than melts at a rate of 50 cm^{3} /min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is
B.