Let f x = a - 1 0 a x a - 1 a x 2 a x a , a ∈ R . Then the sum of the squares of all the values of a for 2 f ' 10 - f ' 5 + 100 = 0 is

Step-by-step solution

Given, f x = a - 1 0 a x a - 1 a x 2 a x a f x = a 1 - 1 0 x a - 1 x 2 a x a = a 1 a 2 + a x + 1 a x + x 2 ⇒ f x = a x + a 2 Now differentiating the function w.r.t x We get, f ' x = 2 a x + a Given, 2 f ' 10 - f ' 5 + 100 = 0 ⇒ 2 × 2 a 10 + a - 2 a 5 + a + 100 = 0 ⇒ 40 a + 4 a 2 - 10 a - 2 a 2 + 100 = 0 2 a 2 + 30 a + 100 = 0 ⇒ a 2 + 15 a + 50 = 0 a + 10 a + 5 = 0 a = - 10 or a = - 5 Required = - 10 2 + - 5 2 = 125