Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
f(b) = pb + 3
`lim_(x -> "b"^+) "f"(x)` = 5b2 − q
∴ pb + 3 = 5b2 − q
∴ p = `(5"b"^2 - "q" - 3)/"b"` is the required condition.
APPEARS IN
संबंधित प्रश्न
Examine the continuity of f(x) = x3 + 2x2 − x − 2 at x = − 2
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)",", "for" x ≠ 2),(= 1/5",", "for" x = 2):}}`at x = 2
Test the continuity of the following function at the point or interval indicated against them:
f(x) `{:( =(x^2 + 8x - 20)/(2x^2 - 9x + 10)",", "for" 0 < x < 3"," x ≠ 2),(= 12",", "for" x = 2),(= (2 - 2x - x^2)/(x - 4)",", "for" 3 ≤ x < 4):}}` at x = 2
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) = `(x^2 - 10x + 21)/(x - 7)`
Discuss the continuity of the following function at the point indicated against them :
f(x) = `{:(=( sqrt(3) - tanx)/(pi - 3x)",", x ≠ pi/3),(= 3/4",", x = pi/3):}} "at" x = pi/3`
The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :
f(x) = `((3 - 8x)/(3 - 2x))^(1/x)`, for x ≠ 0
If f(x) = `(cos^2 x - sin^2 x - 1)/(sqrt(3x^2 + 1) - 1)` for x ≠ 0, is continuous at x = 0 then find f(0)
If f(x) `{:(= (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)",", "for" x ≠ 0), (= "k"",", "for" x = 0):}}` is continuous at x = 0, find k
If f(x) `{:(= (sin2x)/(5x) - "a"",", "for" x > 0),(= 4 ",", "for" x = 0),(= x^2 + "b" - 3",", "for" x < 0):}}` is continuous at x = 0, find a and b
Discuss the continuity of f on its domain, where f(x) `{:(= |x + 1|",", "for" -3 ≤ x ≤ 2),(= |x - 5|",", "for" 2 < x ≤ 7):}`.
Discuss the continuity of f(x) at x = `pi/4` where,
f(x) `{:(= ((sinx + cosx)^3 - 2sqrt(2))/(sin 2x - 1)",", "for" x ≠ pi/4),(= 3/sqrt(2)",", "for" x = pi/4):}`
Show that there is a root for the equation 2x3 − x − 16 = 0 between 2 and 3.
Select the correct answer from the given alternatives:
f(x) = `{:(= (2^(cotx) - 1)/(pi - 2x)",", "for" x ≠ pi/2),(= log sqrt(2)",", "for" x = pi/2):}`
Select the correct answer from the given alternatives:
If f(x) = `(1 - sqrt(2) sinx)/(pi - 4x), "for" x ≠ pi/4` is continuous at x = `pi/4`, then `"f"(pi/4)` =
Select the correct answer from the given alternatives:
If f(x) `{:(= "a"x^2 + "b"x + 1",", "for" |x −1| ≥ 3), (= 4x + 5",", "for" -2 < x < 4):}` is continuous everywhere then,
Select the correct answer from the given alternatives:
f(x) `{:(= ((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))",", "for" x ≠ 0),(= "k"",", "for" x = 0):}` is continuous at x = 0, then ‘k’ =
Select the correct answer from the given alternatives:
If f(x) = `((4 + 5x)/(4 - 7x))^(4/x)`, for x ≠ 0 and f(0) = k, is continuous at x = 0, then k is
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) `{:( = (sin^2pix)/(3(1 - x)^2) ",", "for" x ≠ 1),(= (pi^2sin^2((pix)/2))/(3 + 4cos^2 ((pix)/2)) ",", "for" x = 1):}}` at x = 1
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 + x - 3,"," "for" x ∈ [ -5, -2)),(= x^2 - 5,"," "for" x ∈ (-2, 5]):}`
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
f(x) `{:(= (x^2 + x + 1)/(x + 1)"," , "for" x ∈ [0, 3)),(=(3x +4)/(x^2 - 5)"," , "for" x ∈ [3, 6]):}`
Find a and b if following function is continuous at the point or on the interval indicated against them:
f(x) `{:(= (4tanx + 5sinx)/("a"^x - 1)",", "for" x < 0),(= (9)/(log2)",", "for" x = 0),(= (11x + 7x*cosx)/("b"^x - 1)",", "for" x > 0):}`
Find f(a), if f is continuous at x = a where,
f(x) = `(1 - cos[7(x - pi)])/(5(x - pi)^2`, for x ≠ π at a = π
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) = `{{:((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 the function f(x) = `[tan(π/4 + x)]^(1/x)` for x ≠ 0 is = K for x = 0 continuous at x = 0, then K = ?
If f(x) = `{{:(x, "for" x ≤ 0),(0,
"for" x > 0):}`, then f(x) at x = 0 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 ______.
If f(x) = `{{:((x - 4)/(|x - 4|) + a",", "for" x < 4),(a + b",", "for" x = 4 "is continuous at" x = 4","),((x - 4)/(|x - 4|) + b",", "for" x > 4):}`
then ______.
If f(x) = `{{:((sin^3(sqrt(3)).log(1 + 3x))/((tan^-1 sqrt(x))^2(e^(5sqrt(3)) - 1)x)",", x ≠ 0),( a",", x = 0):}`
is continuous in [0, 1] then a is equal to ______.
If \[\mathrm{f}(x)= \begin{cases} \mathrm{m}x+1, & x\leqslant\frac{\pi}{2} \\ \\ \mathrm{sin}x+\mathrm{n}, & x>\frac{\pi}{2} & \end{cases}\], is continuous at \[x=\frac{\pi}{2},( \begin{array} {c}\mathrm{m,n\in\mathbb{Z}} \end{array})\] then
