Finding the extremum of function y = e ^ x-x-1 Y = the x power of E, minus x, minus 1

Finding the extremum of function y = e ^ x-x-1 Y = the x power of E, minus x, minus 1

y'=e^x-1=0
x=0
At x = 0, y = - 1 is the minimum

Find the extremum of function y = (2e ^ x) + (e ^ - x)
Please be as detailed as possible

Because the function is differentiable everywhere, find the first derivative, Let f '(x) = (2e ^ x) - (e ^ - x) = 0, get the stationary point x = - (LN2 / 2), when x belongs to (negative infinity, - (LN2 / 2)), the first derivative is always less than 0, then the original function monotonically decreases, and when x belongs to (- (LN2 / 2), positive infinity, the first derivative is greater than 0, then the original function monotonically increases, so when x = - (LN2 / 2), there is a minimum value, Ymin = 2 √ 2

Find the extreme point of function. Y = |lg|x-1||
It's OK to draw a picture, as long as you get it

Since y is the absolute value of a function, y ≥ 0,
also
When x = 2 and x = 0, y = 0,
So the function has a minimum
（2,0）（0,0）
There is no maximum
Drawing method
1. Draw an image of y = lgx
2. Move the image right by 1 unit
3. The image below the x-axis is symmetrical with X as the symmetry axis to above x, and the lower part is not retained
4. The whole figure is symmetrical with x = 1 as the symmetry axis, and the figure before and after symmetry is retained

Seeking monotone interval and extremum of function y = x-e ^ x

Y '= 1-e ^ x = 0, the extreme point x = 0
When x

If you want to find the maximum value of the function y = x + 4 / X in the interval [1,3], don't use the derivation, please use the knowledge of senior one,

Let 1 ≤ x1f (x2)
That is, the function f (x) decreases on [1,2]
Similarly, if the function f (x) is increasing on [2,3], then the minimum value of function f (x) on [1,3] is f (2) = 4
In addition, if f (1) = 5 and f (3) = 13 / 3, the maximum value is f (1) = 5

If the image of F (x) = SiNx + acosx is symmetric with respect to the line x = π / 6, what is a?

If x = 0 and x = π / 3 are symmetric with respect to the straight line x = π / 6, they should be equivalent in the analytic expressions,
So sin0 + acos0 = sin π / 3 + ACOS π / 3, that is, 0 + a = root 3 / 2 + A * 1 / 2, so a = root 3

If the image of F (x) = SiNx + acosx is symmetric with respect to x = π / 6, then a=

The image of F (x) = SiNx + acosx is symmetric about x = π / 6
According to the image of cosx
f(x)=mcos(x-π/6)=m(cosxcosπ/6+sinxsinπ/6)
=m(√3/2cosx+1/2sinx)
=m/2sinx+√3m/2cosx
=sinx+acosx
m/2=1,√3m/2=a
m=2,a=√3

If the image of the function y = SiNx + acosx is symmetric with respect to the line x = π / 6, then a=

If x = 0 and x = π / 3, which are equidistant from x = π / 6, then f (0) = f (π / 3), we can get a = radical 3

If f (x) = 2 + SiNx + acosx is symmetric with respect to x = π / 4, then a =?
Answer: a = - 1
Need detailed process. Thank you

Because f (x) is symmetric with respect to x = π / 4
So f (π / 4 + π / 4) = f (π / 4 - π / 4)
That is, f (π / 2) = f (0)
SO 2 + sin (π / 2) + ACOS (π / 2) = 2 + sin0 + acos0
That is, 2 + 1 + A * 0 = 2 + 0 + A * 0
a=1

Given the function f (x) = x + 9 / X (1), judge the monotonicity of F (x) on (0, positive infinity) and prove it. (2) find the definition and range of F (x)

Derivative of F (x) = 1-9 / (square of x)
If the derivative of F (x) is greater than 0, it increases monotonically
Monotonic decreasing less than 0
The domain is x not 0
The range is based on monotonicity
Find the maximum and minimum