Advertisements
Advertisements
प्रश्न
Determine the values of p and q such that the following function is continuous on the entire real number line.
f(x) `{:(= x + 1",", "for" 1 < x < 3),(= x^2 + "p"x + "q"",", "for" |x - 2| ≥ 1):}`
Advertisements
उत्तर
|x – 2| = ± (x – 2) and |x – 2| ≥ 1
∴ x – 2 ≥ 1 or 2 – x ≥ 1
∴ x ≥ 3 or 1 ≥ x i.e., x ≤ 1
∴ the given function is
f(x) `{:(= x^2 + "p"x + "q","," "for" x ≤ 1),(= x + 1, "," "for" 1 < x < 3),(= x^2 + "p"x + "q", "," "for" x ≥ 3):}`
If f is continuous on the entire number line, then it is continuous at x = 1 and x = 3
Continuity at x = 1
Since f is continuous at x = 1,
`lim_(x -> 1^+) "f"(x) = "f"(1)`
∴ `lim_(x -> 1) (x + 1) = (x^2 + "p"x + "q")`
∴ 1 + 1 = 1 + p + q
∴ p + q = 1 ...(1)
Continuity at x = 3
Since f is continuous at x = 3,
`lim_(x -> 3^-) "f"(x)` = f(3)
∴ `lim_(x -> 3) (x + 1) = (x^2 + "p"x + "q")`
∴ 3 + 1 = 9 + 3p + q
∴ 3p + q = – 5
∴ 3p + (1 – p) = – 5 ...[By (1)]
∴ 3p - p = - 5 - 1
∴ 2p = – 6
∴ p = – 3
Substituting p = – 3 in (1), we get,
∴ – 3 + q = 1
∴ q = 1 + 3
∴ q = 4
Hence, p = – 3, q = 4.
APPEARS IN
संबंधित प्रश्न
Examine whether the function is continuous at the points indicated against them:
f(x) = x3 − 2x + 1, for x ≤ 2
= 3x − 2, for x > 2, at x = 2
Examine the continuity of `"f"(x) {:(= sin x",", "for" x ≤ pi/4), (= cos x",", "for" x > pi/4):}} "at" x = pi/4`
Discuss the continuity of the function f(x) = |2x + 3|, at x = `(−3)/(2)`
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))",", "for" x ≠ 2),(= -24",", "for" x = 2):}}` at x = 2
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= ((27 - 2x)^(1/3) - 3)/(9 - 3(243 + 5x)^(1/5))",", "for" x ≠ 0),(= 2",", "for" x = 0):}}` at x = 0.
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) = `(x^2 - 10x + 21)/(x - 7)`
Identify the discontinuity for the following function as either a jump or a removable discontinuity.
f(x) `{:(= x^2 + 3x - 2",", "for" x ≤ 4),(= 5x + 3",", "for" x > 4):}`
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) `{:(= 4 + sin x",", "for" x < pi),(= 3 - cos x",", "for" x > pi):}`
The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :
f(x) `{:(= log_((1 + 3x)) (1 + 5x)",", "for" x > 0),(=(32^x - 1)/(8^x - 1)",", "for" x < 0):}}` at x = 0
The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :
f(x) `{:(= 3x + 2",", "for" -4 ≤ x ≤-2),(= 2x - 3";", "for" -2 < x ≤ 6):}`
For what values of a and b is the function
f(x) `{:(= (x^2 - 4)/(x - 2)",", "for" x < 2),(= "a"x^2 - "b"x + 3",", "for" 2 ≤ x < 3),(= 2x - "a" + "b"",", "for" x ≥ 3):}}` continuous for every x on R?
Show that there is a root for the equation x3 − 3x = 0 between 1 and 2.
Suppose f(x) `{:(= "p"x + 3",", "for" "a" ≤ x ≤ "b"),(= 5x^2 − "q"",", "for" "b" < x ≤ "c"):}`
Find the condition on p, q, so that f(x) is continuous on [a, c], by filling in the blanks.
f(b) = ______
`lim_(x -> "b"^+) "f"(x)` = _______
∴ pb + 3 = _______ − q
∴ p = `"_____"/"b"` is the required condition
Select the correct answer from the given alternatives:
If f(x) = `(12^x - 4^x - 3^x + 1)/(1 - cos 2x)`, for x ≠ 0 is continuous at x = 0 then the value of f(0) is ______.
Select the correct answer from the given alternatives:
If f(x) = [x] for x ∈ (–1, 2) then f is discontinuous at
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= (x^2 - 3x - 10)/(x - 5)",", "for" 3 ≤ x ≤ 6"," x ≠ 5),(= 10",", "for" x = 5),(=(x^2 - 3x - 10)/(x - 5)",", "for" 6 < x ≤ 9):}`
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= 2x^2 - 2x + 5",", "for" 0 ≤ x ≤ 2),(= (1 - 3x - x^2)/(1 - x) "," , "for" 2 < x < 4),(= (x^2 - 25)/(x - 5)",", "for" 4 ≤ x ≤ 7 and x ≠ 5),(= 7",", "for" x = 5):}`
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) = [x + 1] for x ∈ [−2, 2)
Where [*] is greatest integer function.
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
f(x) `{:(= x^2 + 5x + 1"," , "for" 0 ≤ x ≤ 3),(= x^3 + x + 5"," , "for" 3 < x ≤ 6):}`
Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:
f(x) = `((x + 3)(x^2 - 6x + 8))/(x^2 - x - 12)`
Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:
f(x) `{:(= x^2 + 2x + 5"," , "for" x ≤ 3),( = x^3 - 2x^2 - 5",", "for" x > 3):}`
Find k if following function is continuous at the point indicated against them:
f(x) `{:(= ((5x - 8)/(8 - 3x))^(3/(2x - 4))",", "for" x ≠ 2),(= "k"",", "for" x = 2):}}` at x = 2
Find k if following function is continuous at the point indicated against them:
f(x) `{:(= (45^x - 9^x - 5^x + 1)/(("k"^x - 1)(3^x - 1))",", "for" x ≠ 0),(= 2/3",", "for" x = 0):}}` at x = 0
Find a and b if following function is continuous at the point or on the interval indicated against them:
f(x) `{:(= "a"x^2 + "b"x + 1",", "for" |2x - 3| ≥ 2),(= 3x + 2",", "for" 1/2 < x < 5/2):}`
Find f(a), if f is continuous at x = a where,
f(x) = `(1 + cos(pi x))/(pi(1 - x)^2)`, for x ≠ 1 and at a = 1
Solve using intermediate value theorem:
Show that x3 − 5x2 + 3x + 6 = 0 has at least two real root between x = 1 and x = 5
If f(x) = `{((x^4 - 1/81)/(x^3 - 1/27), x ≠ 1/3), (k, x = 1/3):}` is continuous at x = `1/3`, then the value of k is ______
If f(x) = `{:{(tan^-1|x|; "when" x ≠ 0), (pi/4; "when" x = 0):}`, then ______
If f(x) is continuous at x = 3, where
f(x) = ax + 1, for x ≤ 3
= bx + 3, for x > 3 then.
If f(x) = `{{:(tanx/x + secx",", x ≠ 0),(2",", x = 0):}`, then ______.
If f(x) = `{{:((sin5x)/(x^2 + 2x)",", x ≠ 0),(k + 1/2",", x = 0):}` is continuous at x = 0, then the value of k is ______.
Let f be the function defined by
f(x) = `{{:((x^2 - 1)/(x^2 - 2|x - 1| - 1)",", x ≠ 1),(1/2",", x = 1):}`
If f(x) = `1/(1 - x)`, the number of points of discontinuity of f{f[f(x)]} is ______.
If f(x) = `{{:(log(sec^2 x)^(cot^2x)",", "for" x ≠ 0),(K",", "for" x = 0):}`
is continuous at x = 0, then K is ______.
If the function f(x) defined by
f(x) = `{{:(x sin 1/x",", "for" x = 0),(k",", "for" x = 0):}`
is continuous at x = 0, then k is equal to ______.
