Advertisements
Advertisements
प्रश्न
f(x) = xx has a stationary point at ______.
पर्याय
x = e
x = `1/"e"`
x = 1
x = `sqrt("e")`
Advertisements
उत्तर
f(x) = xx has a stationary point at x = `1/"e"`.
Explanation:
We have f(x) = xx
Taking log of both sides, we have
log f(x) = x log x
Differentiating both sides w.r.t. x, we get
`1/("f"(x)) * "f'"(x) = x * 1/x + log x * 1`
⇒ f'(x) = f(x)[1 + log x] = xx[1 + log x]
To find stationary point, f'(x) = 0
∴ xx[1 + log x] = 0
xx ≠ 0 ∴ 1 + log x = 0
⇒ log x = – 1
⇒ x = e–1
⇒ x = `1/"e"`
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles x2ex ?
Differentiate tan (x° + 45°) ?
Differentiate `2^(x^3)` ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate (log sin x)2 ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
If \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta = \frac{\pi}{2}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate (log x)x with respect to log x ?
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\] then `f' (x)` is equal to ____________ .
Find the second order derivatives of the following function log (log x) ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If y = a xn + 1 + bx−n and \[x^2 \frac{d^2 y}{d x^2} = \lambda y\] then write the value of λ ?
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
If p, q, r, s are real number and pr = 2(q + s) then for the equation x2 + px + q = 0 and x2 + rx + s = 0 which of the following statement is true?
