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) = sqrt(x) - x/3, x ∈ [0, 9]`
उत्तर
f(x) = `sqrt(x) - x/3, x ∈ [0, 9]`
f(0) = 0, f(9) = `sqrt(9) - 9/3` = 3 – 3 = 0
⇒ f(0) = 0 = f(9)
f(x) is continuous on [0, 9]
f(x) is differentiable on (0, 9)
Now f'(x) = `1/(2sqrt(x)) - 1/3`
Since, the tangent is parallel to x-axis.
f'(x) = 0
`1/(2sqrt(x)) - 1/3` = 0
⇒ `1/(2sqrt(x)) = 1/3`
`sqrt(x) = 3/2`
x = `9/4`
x = `9/4 ∈ (0, 9)`
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)` = 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) = (x^2 - 2x)/(x + 2), x ∈ [-1, 6]`
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]`
Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:
`f(x) = |3x + 1|, x ∈ [-1, 3]`
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]`
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 - 2)(x - 7), x ∈ [3, 11]`
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`
Suppose that for a function f(x), f'(x) ≤ 1 for all 1 ≤ x ≤ 4. Show that f(4) – f(1) ≤ 3
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
Show that there lies a point on the curve `f(x) = x(x + 3)e^(pi/2), -3 ≤ x ≤ 0` where tangent drawn is parallel to the x-axis
Choose the correct alternative:
The number given by the Mean value theorem for the function `1/x`, x ∈ [1, 9] is
