Advertisements
Advertisements
प्रश्न
Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:
`f(x) = (x^2 - 2x)/(x + 2), x ∈ [-1, 6]`
उत्तर
f(– 1) = `(1 + 2)/(-1 + 2)` = 3
f(6) = `(36 - 12)/8 = 24/8` = 3
⇒ f(– 1) = 3 = f(6)
f(x) is continuous on [– 1, 6]
f(x) is differentiable on (– 1, 6)
Now, f'(x ) = `((x + 2)(2x - 2) - (x^2 - 2x)(1))/(x + 2)^2`
= `(x^2 + 4x - 4)/(x + 2)^2`
Since the tangent is parallel to the x-axis.
f'(x) = 0
⇒ x2 + 4x – 4 = 0
⇒ x = `- (4 +- sqrt(16 + 16))/2`
x = `- (4 +- 4sqrt(2))/2`
= `- 2 +- 2sqrt(2)`
x = `- 2 +- 2sqrt(2) ∈ (-1, 6)`
APPEARS IN
संबंधित प्रश्न
Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals
`f(x) = |1/x|, x ∈ [- 1, 1]`
Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals
`f(x)` = tan x, x ∈ [0, π]
Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals
`f(x)` = x – 2 log x, x ∈ [2, 7]
Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:
`f(x)` = x2 – x, x ∈ [0, 1]
Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:
`f(x) = sqrt(x) - x/3, x ∈ [0, 9]`
Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:
`f(x) = (x + 1)/x, x ∈ [-1, 2]`
Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:
`f(x) = x^3 - 3x + 2, x ∈ [-2, 2]`
Show that the value in the conclusion of the mean value theorem for `f(x) = 1/x` on a closed interval of positive numbers [a, b] is `sqrt("ab")`
Show that the value in the conclusion of the mean value theorem for `f(x) = "A"x^2 + "B"x + "C"` on any interval [a, b] is `("a" + "b")/2`
A race car driver is kilometer stone 20. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours
Does there exist a differentiable function f(x) such that f(0) = – 1, f(2) = 4 and f(x) ≤ 2 for all x. Justify your answer
Using Mean Value Theorem prove that for, a > 0, b > 0, |e–a – e–b| < |a – b|
Choose the correct alternative:
The number given by the Rolle’s theorem for the function x3 – 3x2, x ∈ [0, 3] is
Choose the correct alternative:
The number given by the Mean value theorem for the function `1/x`, x ∈ [1, 9] is
