Verify Rolle’s theorem for the following function:
on [0, ]
Given equation is,
(i) f(x) is continuous on [0, ] because are continuous function on its domain.
(ii) e- x and sin(x) are diffentiable on (0, )
(iii) f(0) = e- 0.sin(0) = 0
And, f() = e- .sin() = 0
(iv) Let c be number such that f'(c) = 0
e- c (cos(c) - sin(c)) = 0
Therefore, Rolle's theorem is verified.
Find the points on the curve 4x3 - 3x + 5 at which the equation of the tangent is parallel to the x-axis.
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.