Given the function f (x) = inx-x square + ax, obtain the extreme value at x = 1, find the value of real number a, and find the monotone interval of function f (x) Why do you give different answers? It's two questions. Give a detailed and correct answer, please

Given the function f (x) = inx-x square + ax, obtain the extreme value at x = 1, find the value of real number a, and find the monotone interval of function f (x) Why do you give different answers? It's two questions. Give a detailed and correct answer, please

If the extreme value is obtained at x = 1, its derivative is 0 at x = 1
f'(x)=1/x-2x+a
f'(1)=1-2+a=0
a=1
F '(x) = 1 / x-2x + 1 > 0 is monotonically increasing
∵x>0∴1-2x^2+x>0 -1/2

Let function f (x) = INX ax. Find the extreme point of function f (x)

Function definition field: x > 0
Let f '(x) = 1 / x-a = 0
If a ≤ 0, there is no extreme value;
If a > 0 and x = 1 / A, take the extreme value, f (1 / a) = - LNA-1

Let f (x) = inx-ax find the extreme point of function f (x). When a > 0, there is always f (x)

Firstly, it can be determined that the value range of X is (0, + infinity), and the derivative function = 1 / x-a,
Discuss the sign of derivative function,
(1) When a is less than or equal to 0, the derivative function is always greater than zero. At this time, the function f (x) is an increasing function and has no extreme value in the definition domain
(2) When a > 0, when 1 / x-a = 0, that is, when x = 1 / A, the function f (x) obtains the extreme value
If f (x) is constant when a > 0

Given that f (x) = ln (2x + 1), if the derivative of F (x) + F (x) = a has a solution, find the value range of A

The derivative of F (x) + F (x) = a gets ln (2x + 1) + 2 / (2x + 1) = a, let g (x) = ln (2x + 1) + 2 / (2x + 1), the derivative of G (x) increases at (1 / 2, + OO), and f (x) = ln (2x + 1). If the derivative of F (x) + F (x) = a has a solution, substitute 1 / 2 to get a > = LN2 + 1

What is the derivative of Ln (x + 1)

y=ln(x+1)
Let x + 1 = t
y=lnt
y'=(lnt)'*t'
y'=1/(x+1)

Why is the derivative of Ln (2x + 3) 2 / (2x + 3) Not 2 / x?

This is a composite function
lnx ' =1/x
ln（2x+3） =1/2x+3 （2x+3）'
=2/2x+3