Question

find the function f given that satisfies f (x)=36x^2 +24x and its graph has a horizontal tangent lies at the point

Answer 1

Is the f (x) supposed to be the derivative? And what’s the point where the tangent is supposed to be?

Answer 2

Is this your full question?

Find the function f given that it satisfies f''(x)=36x^2+24x
and its graph has a horizontal tangent line at the point (0,1)

if so, you'd take the integral once to get the first derivative. you should have a + C term since this is an indefinite integral. They give you the horizontal tangent at (0,1) meaning that the first derivative is equal to 0 when x = 0. plug in and solve for the C.

take the integral again to get the original function f. this time you should have a Cx and a + D term. because the function has a tangent at (0,1) it must also be defined at (0,1) so you can plug in this point into the function along with the C-value from before to get the final constant D. from there, re-write the whole function f with the appropriate C and D value substituted.