Advertisements
Advertisements
प्रश्न
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
उत्तर
f(x) = `x(x + 3)e^(pi/2), -3 ≤ x ≤ 0`
f(x) is continuous in [– 3, 0]
f(x) is differentiable in (– 3, 0)
f(– 3) = `-3(-3 + 3)"e"^((-pi)/2)` = 0
f(0) = 0
⇒ f(– 3) = f(0) = 0
Since the tangent is parallel to x-axis.
f'(c) = 0
`"e"^((- pi)/2) (2"c" + 3) = 0 `
Where f'(x) = `"e"^((- pi)/2) (2x + 3)`
2c + 3 = 0
c = `- 3/2 ∈ (-3, 0)`
∴ The point lies on the curve.
APPEARS IN
संबंधित प्रश्न
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]`
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")`
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
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
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
