Question

If a function f(x) defined by

f(x) = `{{:("ae"^x + "be"^-x",", -1 ≤ x < 1),("c"x^2",", 1 ≤ x ≤ 3),("a"x^2 + 2"c"x",", 3 < x ≤ 4):}`

be continuous for some a, b, c ε R and f'(0) + f'(2) = e, then the value of a is ______.

पर्याय

• `1/("e"^2 - 3"e" + 13)`

• `"e"/("e"^2 - 3"e" - 13)`

• `"e"/("e"^2 + 3"e" + 13)`

• `"e"/("e"^2 - 3"e" + 13)`

Answer 1

If a function f(x) defined by

f(x) = `{{:("ae"^x + "be"^-x",", -1 ≤ x < 1),("c"x^2",", 1 ≤ x ≤ 3),("a"x^2 + 2"c"x",", 3 < x ≤ 4):}`

be continuous for some a, b, c ε R and f'(0) + f'(2) = e, then the value of a is `underlinebb(e/(e^2 - 3e + 13))`.

Explanation:

f(x) = `{{:("ae"^x + "be"^-x",", -1 ≤ x < 1),("c"x^2",", 1 ≤ x ≤ 3),("a"x^2 + 2"c"x",", 3 < x ≤ 4):}`

Continuous at x = 1

LHL = RHL

ae¹ + be^–1 = c

⇒ `"ae" + "b"/"e"` = c

⇒ ae² + b = ec  ...(i)

Cont at x = 3

LHL = RHL

9c = 9a + 3c × 2

3c = 9a

c = 3a  ...(ii)

f'(0) + f'(2) = e  ...(Given)

{ae^x + b)e^–x)(–1)}_x=0 + {2cx}_x=2 = e

a – b + 2c2 = e

a – b + 4c = e  ...(iii)

a – (ec – ae²) + 4c = e   from (i)

a – (3ae – ae²) + 12a = e  from (ii)

a{1 – 3e + e² + 12} = e

a = `"e"/("e"^2 - 3"e" + 13)`