Question

If a function defined by f(x) = `{(kx + 1","  x ≤ π),(cosx","  x > π):}` is continuous at x = π, then the value of k is ______.

पर्याय

• π

• `(-1)/π`

• 0

• `(-2)/π`

Answer 1

If a function defined by `f(x) = {(kx + 1",", x ≤ π),(cosx",", x > π):}` is continuous at x = π, then the value of k is `underlinebb((-2)/π)`.

Explanation:

f(x) is continuous at x = π

If L.H.L. = R.H.L. = f(π)

i.e., `lim_(x -> π^-) f(x) = lim_(x -> π^+) f(x) = f(π)`

L.H.L. at x → π

`lim_(x -> π^-) f(x) = lim_(h -> 0) f(π - h)`

= `lim_(h -> 0) k(π - h) + 1`

= k(π − 0) + 1

= kπ + 1

R.H.L. at x → π

`lim_(x -> π^+) f(x) = lim_(h -> 0) f(π + h)`

= `lim_(h -> 0) cos (π + h)`

= cos (π + 0)

= cos (π)

= −1

Since L.H.L. = R.H.L.

kπ + 1 = −1

kπ = −2

k = `(-2)/π`