Application of Derivatives

Let f be a differentiable function satisfying f(x) = 1 - 2x + 0x e(x-t) f(t) dt, x R and let g(x) = 0x (f(t) + 2)15(t - 4)6(t + 12)17 dt, x R. If p and q are respectively the points of local minima and local maxima of g, then the value of |p + q| is equal to .

Let (2 , ) be the largest interval in which the function f(t) = |t+1|t2, t < 0, is strictly decreasing. Then the local maximum value of the function g(x) = 2 e(x-2) + x2 + 4x - , x > 2, is

Let f : R R be a twice differentiable function such that the quadratic equation f(x)m2 - 2f'(x)m + f''(x) = 0 in m, has two equal roots for every x R. If f(0) = 1, f'(0) = 2, and ( , ) is the largest interval in which the function f( e x - x) is increasing, then + is equal to .

If the set of all values of a, for which the equation 5x3-15x-a=0 has three distinct real roots, is the interval ( , ), then -2 is equal to .

Let the set of all positive values of , for which the point of local minimum of the function (1 + x ( 2 - x2 ) ) satisfies x2 + x + 2x2 + 5x + 6 < 0, be ( , ). Then 2 + 2 is equal to .

Let the set of all values of p, for which f(x) = (p2 - 6p + 8 ) ( 2 2x - 2 2x ) + 2(2 - p)x + 7 does not have any critical point, be the interval (a, b). Then 16ab is equal to .

Let A be the region enclosed by the parabola y2 = 2x and the line x = 24. Then the maximum area of the rectangle inscribed in the region A is .

Let f : R R be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2 and f(4) = -2. Then, the minimum number of zeros of (3f'f'' + ff''')(x) is .

Let S=(-1, ) and f : S R be defined as f(x)= -1x (et-1 )11(2t-1)5(t-2)7(t-3)12(2t-10)61dt, Let p= Sum of squares of the values of x, where f(x) attains local maxima on S, and q= Sum of the values of x, where f(x) attains local minima on S. Then, the value of p2+2q is .

Let f(x)=2x-x2,x R. If m and n are respectively the number of points at which the curves y=f(x) and y=f'(x) intersect the x-axis, then the value of m+n is .