Advertisements
Advertisements
प्रश्न
Discuss the continuity and differentiability of f(x) = (2x + 3) |2x + 3| at x = `- 3/2`
उत्तर
If `x ≥ -3/2`, |2x + 3| = 2x + 3 and if `x < -3/2`, |2x + 3| = − (2x + 3)
∴ f(x) `{:(= (2x + 3)^2"," , "for" x ≥ - 3/2),(= -(2x + 3)^2"," , "for" x < - 3/2):}`
`"R" "f'"(-3/2) = lim_("h" -> 0) ("f"(- 3/2 + "h") - "f"(-3/2))/"h"`
= `lim_("h" -> 0) ([2(- 3/2 + "h") + 3]^2 - [2(- 3/2) + 3]^2)/"h" ...[because "f"(x) = (2x + 3)^2"," "for" x ≥ - 3/2]`
= `lim_("h" -> 0) ([(-3 + 2"h") + 3]^2 - 0)/"h"`
= `lim_("h" -> 0) (4"h"^2)/"h"`
= `lim_("h" -> 0) (4"h")` ...[∵ h → 0 ∴ h ≠ 0]
= 0
Similarly, `"L" "f'"(- 3/2)` = 0
∴ `"R""f'"(- 3/2) = "L" "f'"(- 3/2)` = 0
∴ f is differentiable at x = `- 3/2` and hence continuous at x = `-3/2`.
APPEARS IN
संबंधित प्रश्न
Find the derivative of the following function w.r.t. x.:
x–9
Find the derivative of the following functions w. r. t. x.:
`x^(3/2)`
Find the derivative of the following function w. r. t. x.:
`7xsqrt x`
Differentiate the following w. r. t. x.: x5 + 3x4
Differentiate the following w. r. t. x. : `x sqrtx + logx − e^x`
Differentiate the following w. r. t. x. : `2/7 x^(7/2) + 5/2 x^(2/5)`
Differentiate the following w. r. t. x. : x3 log x
Differentiate the following w. r. t. x. : ex log x
Differentiate the following w. r. t. x. : x3 .3x
Find the derivative of the following w. r. t. x by using method of first principle:
x2 + 3x – 1
Find the derivative of the following w. r. t. x by using method of first principle:
sin (3x)
Find the derivative of the following w. r. t. x by using method of first principle:
e2x+1
Find the derivative of the following w. r. t. x by using method of first principle:
3x
Find the derivative of the following w. r. t. x by using method of first principle:
log (2x + 5)
Find the derivative of the following w. r. t. x by using method of first principle:
tan (2x + 3)
Find the derivative of the following w. r. t. x by using method of first principle:
`x sqrt(x)`
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
`2^(3x + 1)` at x = 2
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
`"e"^(3x - 4)` at x = 2
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
cos x at x = `(5pi)/4`
Show that the function f is not differentiable at x = −3, where f(x) `{:(= x^2 + 2, "for" x < - 3),(= 2 - 3x, "for" x ≥ - 3):}`
Show that f(x) = x2 is continuous and differentiable at x = 0
Discuss the continuity and differentiability of f(x) = x |x| at x = 0
Discuss the continuity and differentiability of f(x) at x = 2
f(x) = [x] if x ∈ [0, 4). [where [*] is a greatest integer (floor) function]
Test the continuity and differentiability of f(x) `{:(= 3 x + 2, "if" x > 2),(= 12 - x^2, "if" x ≤ 2):}}` at x = 2
Select the correct answer from the given alternative:
If y = `("a"x + "b")/("c"x + "d")`, then `("d"y)/("d"x)` =
Select the correct answer from the given alternative:
If f(x) `{:(= 2x + 6, "for" 0 ≤ x ≤ 2),(= "a"x^2 + "b"x, "for" 2 < x ≤4):}` is differentiable at x = 2 then the values of a and b are
Select the correct answer from the given alternative:
If f(x) `{:( = x^2 + sin x + 1, "for" x ≤ 0),(= x^2 - 2x + 1, "for" x ≤ 0):}` then
Select the correct answer from the given alternative:
If, f(x) = `x^50/50 + x^49/49 + x^48/48 + .... +x^2/2 + x + 1`, thef f'(1) =
Find the values of p and q that make function f(x) differentiable everywhere on R
f(x) `{:( = 3 - x"," , "for" x < 1),(= "p"x^2 + "q"x",", "for" x ≥ 1):}`
Determine the values of p and q that make the function f(x) differentiable on R where
f(x) `{:( = "p"x^3",", "for" x < 2),(= x^2 + "q"",", "for" x ≥ 2):}`
Determine all real values of p and q that ensure the function
f(x) `{:( = "p"x + "q"",", "for" x ≤ 1),(= tan ((pix)/4)",", "for" 1 < x < 2):}` is differentiable at x = 1
