Question 1

Let f(x) and g(x) be twice differentiable functions satisfying f''(x) = g''(x) for all x R , f'(1) = 2g'(1) = 4 and g(2) = 3f(2) = 9. Then f(25) - g(25) is equal to :

Accepted Answer

Given f''(x) = g''(x) for all x R . Let h(x) = f(x) - g(x). Taking the second derivative, h''(x) = f''(x) - g''(x) = 0. Integrating with respect to x, h'(x) = c_1. Given f'(1) = 4 and g'(1) = 2, h'(1) = f'(1) - g'(1) = 4 - 2 = 2. Therefore, c_1 = 2, which gives h'(x) = 2. Integrating again with respect to x, h(x) = 2x + c_2. Given 3f(2) = 9 f(2) = 3 and g(2) = 9, h(2) = f(2) - g(2) = 3 - 9 = -6. Substituting x = 2 in h(x), h(2) = 2(2) + c_2 = -6 c_2 = -10. Thus, h(x) = 2x - 10. Substituting x =

Question 2

Let f be a real polynomial of degree n such that f(x) = f'(x) f''(x), for all x R . If f(0) = 0, then 36 (f'(2) + f''(2) + _0^2 f(x)\,dx ) is equal to:

Accepted Answer

Let the degree of the polynomial f(x) be n. The degree of f'(x) is n-1 and the degree of f''(x) is n-2. Since f(x) = f'(x) f''(x), equating the degrees on both sides gives: n = (n-1) + (n-2) n = 3 Let f(x) = ax^3 + bx^2 + cx + d. Given f(0) = 0, we get d = 0. Thus, f(x) = ax^3 + bx^2 + cx. Differentiating f(x) with respect to x: f'(x) = 3ax^2 + 2bx + c f''(x) = 6ax + 2b Substituting these into the given equation f(x) = f'(x) f''(x): ax^3 + bx^2 + cx = (3ax^2 + 2bx + c)(6ax + 2b) ax^3 + bx^2 + cx

Question 3

Let f(x)=x^ 3 +x^ 2 f^ (1)+2 x f^ (2)+f^ (3), x R . Then the value of f^ (5) is:

Accepted Answer

Let a = f'(1), b = f''(2), c = f'''(3). Then f(x) = x^3 + ax^2 + 2bx + c. Taking derivatives: f'(x) = 3x^2 + 2ax + 2b, f''(x) = 6x + 2a, f'''(x) = 6. From f'(1) = a: 3 + 2a + 2b = a gives a = -3 - 2b. From f''(2) = b: 12 + 2a = b. From f'''(3) = c: c = 6. Substituting the first into the second: b = 12 + 2(-3 - 2b) = 6 - 4b gives 5b = 6, so b = 6 5 . Then a = -3 - 12 5 = - 27 5 . f(x) = x^3 - 27 5 x^2 + 12 5 x + 6 f'(x) = 3x^2 - 54 5 x + 12 5 f'(5) = 75 - 54 + 12 5 = 117 5

Differentiation — Previous Year Questions & Solutions