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Let f (x) = x4 – 4x, then
- a)f is increasing in [−∞,1)
- b)f is decreasing in [1,∞)
- c)f is increasing in [1,∞)
- d)none of these.
Correct answer is option 'C'. Can you explain this answer?
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Let f (x) =x4– 4x, thena)f is increasing in[−∞,1)b)f...
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Let f (x) =x4– 4x, thena)f is increasing in[−∞,1)b)f...
Explanation:
Increasing and decreasing functions:
An increasing function is one in which the value of the function increases as the input increases, while a decreasing function is one in which the value of the function decreases as the input increases.
Finding the critical points:
To determine where the function f(x) = x^4 - 4x is increasing or decreasing, we first need to find the critical points. Critical points occur where the derivative of the function is equal to zero or undefined.
Finding the derivative:
First, find the derivative of f(x) with respect to x:
f'(x) = 4x^3 - 4
Finding critical points:
Next, set the derivative equal to zero and solve for x:
4x^3 - 4 = 0
4x^3 = 4
x^3 = 1
x = 1
Testing intervals:
Now, we can test the intervals around the critical point x = 1 to determine where f(x) is increasing or decreasing.
Testing interval [0,1):
Choose x = 0.5 (0 < x="" />< />
f'(0.5) = 4(0.5)^3 - 4 = -3 < />
Since the derivative is negative in this interval, f(x) is decreasing in [0,1).
Testing interval (1,∞):
Choose x = 2 (x > 1)
f'(2) = 4(2)^3 - 4 = 28 > 0
Since the derivative is positive in this interval, f(x) is increasing in (1,∞).
Conclusion:
Therefore, the function f(x) = x^4 - 4x is increasing in the interval [1,∞).
Increasing and decreasing functions:
An increasing function is one in which the value of the function increases as the input increases, while a decreasing function is one in which the value of the function decreases as the input increases.
Finding the critical points:
To determine where the function f(x) = x^4 - 4x is increasing or decreasing, we first need to find the critical points. Critical points occur where the derivative of the function is equal to zero or undefined.
Finding the derivative:
First, find the derivative of f(x) with respect to x:
f'(x) = 4x^3 - 4
Finding critical points:
Next, set the derivative equal to zero and solve for x:
4x^3 - 4 = 0
4x^3 = 4
x^3 = 1
x = 1
Testing intervals:
Now, we can test the intervals around the critical point x = 1 to determine where f(x) is increasing or decreasing.
Testing interval [0,1):
Choose x = 0.5 (0 < x="" />< />
f'(0.5) = 4(0.5)^3 - 4 = -3 < />
Since the derivative is negative in this interval, f(x) is decreasing in [0,1).
Testing interval (1,∞):
Choose x = 2 (x > 1)
f'(2) = 4(2)^3 - 4 = 28 > 0
Since the derivative is positive in this interval, f(x) is increasing in (1,∞).
Conclusion:
Therefore, the function f(x) = x^4 - 4x is increasing in the interval [1,∞).
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