# [SOLVED] Find a function f such that f '(x) = 3x^3 and the line 81x + y = 0 is tangent to the graph of f. f(x) =?

Find a function f such that f '(x) = 3x^3 and the line 81x + y = 0 is tangent to the graph of f. f(x) =?

by
selected by

## The Best answer for Find a function f such that f '(x) = 3x^3 and the line 81x + y = 0 is tangent to the graph of f. f(x) =?

f '(x) = 3x^3 = -81 => x = -3

y - f(-3) = -81(x + 3)

y = -81x - 243 + f(-3)

f(-3) - 243 = 0 => f(-3) = 243

f(x) = 3/4*x^4 + C

243 = 243/4 + C => C = 182.25

f(x) = 3/4*x^4 + 182.25

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat.
by

If f'(x) = 3x^3 then f(x) = (3/4)x^4 + c, for some value of c. Since 81x + y = 0 is a tangent then the gradient of this line = -81 = f'(x) = 3x^3 (the dash on f'(x) isn't showing up very well). This means that 3x^3 = - 81, so x^3 = -27, so x = -3. Thereforethe point of contact (since it lies on the line) is

(-3,243) and this point is also on the curve, so (3/4)(-3)^4 +c = 243; this reduces to (243/4) + c = 243

so c = 243 - 243/4 = (3/4)243 , so the equation of the curve is y = (3/4)x + (3/4)243

which might be written y = (3/4)(x + 243) or 4y = 3(x + 243) or 3x + 729

This is the Correct Answer for Find a function f such that f '(x) = 3x^3 and the line 81x + y = 0 is tangent to the graph of f. f(x) =?

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat.

f '(x) = 3x^3

81 x + y = 0

so y = 81 x

y' = 81

Let 3x^3 = 81

x^3 = 27

x = 3

when x = 3 ,

81x + y = 0

y = -81x = -81(3) = -243

ie ( (3 , -243)

f ' (x) = 3x^3

so f(x) = 3/4 x^4 + c

-243 = 3/4 (3)^4 +c

-243 = 3/4 *81 +c

c = -243 - 243 / 4

= - 972/4 - 243/4

= -1215/ 4

so f(x) = 3/4 x^4 - 1215/4

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat.

anti derivative is (3x^4)/4 +C.

y=-81x.

m=-81

-81=3x^3

=-3

y=-81(3) =243

243=3/4(-3)^4 +C

C=729/4

The final answer of the problem is f(x)=(3x^4)/4 +729/4

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat.