Question

F(x)=exX⁴, f'(x)>0 реши

Answer 1

f(x)=eˣх⁴, f'(x)>0

f'(x)=eˣх⁴+4х³eˣ=х³еˣ*(х+4)

х³еˣ*(х+4)>0, еˣ>0 для любого х

__-4________0_____

+              -             +

х∈(-∞;-4)∪(0;+∞)

Answer 2

To solve the inequality f'(x) > 0 for the given function f(x) = e^x * X^4, we need to find the critical points of the function and determine the intervals where the derivative is positive.

Step 1: Find the derivative of the function f(x) with respect to x: f'(x) = d/dx (e^x * X^4) f'(x) = e^x * d/dx (X^4) f'(x) = e^x * 4X^3

Step 2: Set the derivative equal to zero to find the critical points: e^x * 4X^3 = 0

Since e^x is always positive and nonzero, we can divide both sides by e^x: 4X^3 = 0

Step 3: Solve for X: X = 0

Step 4: Determine the intervals where the derivative is positive: Now, we'll examine the intervals around the critical point X = 0. We can choose any value in these intervals and plug them into the derivative e^x * 4X^3 to determine its sign.

1. For X < 0: Choose X = -1 (any negative value) f'(-1) = e^(-1) * 4(-1)^3 = -4e^(-1) < 0

2. For 0 < X < 1: Choose X = 1/2 (any positive value less than 1) f'(1/2) = e^(1/2) * 4(1/2)^3 = e^(1/2) > 0

3. For X > 1: Choose X = 2 (any positive value greater than 1) f'(2) = e^2 * 4(2)^3 = 32e^2 > 0

Step 5: Final solution: The derivative f'(x) is greater than zero (f'(x) > 0) in the intervals (-∞, 0) and (1, ∞). In mathematical notation, the solution is:

(-∞, 0) U (1, ∞)

This means the function f(x) = e^x * X^4 is increasing in the intervals (-∞, 0) and (1, ∞).