Advertisements
Advertisements
प्रश्न
Discuss extreme values of the function f(x) = x.logx
उत्तर
f(x) = x.logx
Differentiating w.r.t. x,
`f'(x) = x . 1/x + logx.1`
f'(x) = 1 + logx
Differentiating again w.r.t. x,
`f''(x) = 1/x`
For maxima or minima,
f'(x) = 0
∴ 1 + logx = 0
∴ logx = -1
∴ x = `e^-1`
∴ `f''(1/e) = 1/(1/e)`
∴ `f''(1/e) = e`
∴ `f''(1/e) > 0`
∴ f(x) is minimum at x = `1/e`
APPEARS IN
संबंधित प्रश्न
If y=eax ,show that `xdy/dx=ylogy`
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `dy/dx` in the following:
2x + 3y = sin y
Find `dy/dx` in the following:
sin2 y + cos xy = k
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Show that the derivative of the function f given by
Is |sin x| differentiable? What about cos |x|?
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Differentiate e4x + 5 w.r..t.e3x
Differentiate tan-1 (cot 2x) w.r.t.x.
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"`, if : x = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.
Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
If x2 + 6xy + y2 = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
Find the nth derivative of the following:
`(1)/x`
Choose the correct option from the given alternatives :
If y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to
Choose the correct option from the given alternatives :
If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are
Differentiate the following w.r.t. x : `sin^2[cot^-1(sqrt((1 + x)/(1 - x)))]`
If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.
DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 - x^2))/(1 - 2x^2)))`.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
Find `"dy"/"dx"` if, xy = log (xy)
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
Choose the correct alternative.
If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ?
If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.
Find `dy/dx if, x= e^(3t), y = e^sqrtt`
If log(x+y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx` if , x = `e^(3t), y = e^(sqrtt)`
If log (x+y) = log (xy) + a then show that, `dy/dx= (-y^2)/(x^2)`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
If log(x + y) = log(xy) + a then show that, `dy/dx = (−y^2)/x^2`
If log(x + y) = log(xy) + a then show that, `dy/dx=(-y^2)/x^2`
Find `dy/(dx) "if" , x = e^(3t), y = e^sqrtt`.
