Question

1)sin(п/2-х)=sin(-п/4) 2)f(x)=2^2*e^x 3)f(x) =2x^3 - 1/2x^4 - 8

Answer 1

1)Sin(п/2-х)=Sin(-п/4)
Cosx = -√2/2
х = +-arcCos(-√2/2) + 2πk, k ∈ Z
x = +- 3π/4 + 2πk , k ∈Z
2)f(x)=2^2*e^x = 4eˣ             (есть формула (eˣ)' = eˣ)
f'(x) = 4eˣ
3)f(x) =2x^3 - 1/2x^4 - 8  (есть формула, по которой: (х²)' = 2x;  (x³)' = 3x²;(x⁴)' = 4x³ и т.д.)
f'(x) = 6x² -2x³

Answer 2

Let's go through each of the equations one by one:

1. sin(π/2 - x) = sin(-π/4)

We can use the trigonometric identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b). In this case, a = π/2 and b = x. Therefore:

sin(π/2 - x) = sin(π/2)cos(x) - cos(π/2)sin(x) 1 * cos(x) - 0 * sin(x) = cos(x)

Now, we need to find the value of sin(-π/4):

sin(-π/4) = -sin(π/4) = -1/√2

So, the equation becomes:

cos(x) = -1/√2

To solve for x, we can take the inverse cosine (arccos) of both sides:

x = arccos(-1/√2)

The solution for x will depend on the range you're considering (usually -π ≤ x ≤ π or 0 ≤ x ≤ 2π). You can use a calculator to find the numerical value of x in the desired range.

2. f(x) = 2^2 * e^x

Simplifying the equation, we have:

f(x) = 4 * e^x

This equation represents an exponential function with a base of e. It will increase rapidly as x increases.

3. f(x) = 2x^3 - (1/2)x^4 - 8

This is a polynomial function. It is a cubic function with a quartic term.

The function can be simplified as:

f(x) = 2x^3 - (1/2)x^4 - 8

There is not much more we can do with this equation without specific instructions or constraints. If you have a specific task or question related to this function, please let me know, and I'll be happy to assist you further.