NTA Abhyas JEE Main2020MathematicsContinuity and DifferentiabilityPractice

If the function f x = a x + 7 : 0 ≤ x < 9 b x + 5 : x ≥ 9 is differentiable for x ≥ 0 , then the value of 5 a + 6 b is equal to

Options

A240 23
B10
C80 23
D250 23

Correct answer

B. 10

Step-by-step solution

f x must be continuous and differentiable at x = 9 For continuity at x = 9 f 9 - = f 9 + = f 9 ⇒ a 9 + 7 = b 9 + 5 ⇒ 4 a = 9 b + 5 …. 1 For differentiability at x = 9 f ' x = a 2 x + 7 : 0 ≤ x < 9 b : x ≥ 9 f ′ 9 − = f ′ 9 + a 2 9 + 7 = b ⇒ a = 8 b … 2 From ( 1 ) ,   32 b = 9 b + 5 ⇒ b = 5 23 Hence, 5 a + 6 b = 40 b + 6 b = 46 b = 1 0

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