Solve the following system of linear equation using matrix method.
3x + y + z = 1, 2x + 2z = 0, 5x + y + 2z = 2.
Verify the conditions of Rolle's Theorem for the following function.
Find a point in the interval, where the tangent to the curve is parallel to x - axis.
Since logarithmic function is differentiable and continuous and so continuous on its domain.Therefore, f(x) is continuous on [- 1, 1] and differentiable on [1, - 1].
Given,
f(- 1) = log((- 1)2 + 2) - log(3)
= log(3) - log(3) = 0
f(1) = log((1)2 + 2) - log(3)
= log(3) - log(3) = 0
Also, f(0) = 0
Thus, f(- 1) = f(1) = 0
All the conditions of Rolle's theorem are satisfied then, there exists 'c' in (- 1, 1) such that f'(c) = 0
Therefore, f'(x) = 0
Put x = c in above equation, we get
f'(c) =
Thus, c = 0 lies between - 1 and 1. Hence, Rolle's theorem is verified.
The point where the tangent is parallel to x axis is (0, ).
Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose legth of latus rectum is 10. Also, find its eccentricity.
A rectangle is inscribed in semicircle of ardius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area.