Increasing And Decreasing Functions

Step 1: f'(x)=²x-4
Step 2: ²x<4 ²x>\frac14 |x|<(π)/(3)

Step 1: Differentiate. f'(x)=cos x-k
Step 2: Condition for decrease. f'(x)\le0 cos x k
Step 3: Conclusion. For all x, this holds when k\ge1.

Step 1: Analyze the function.
For the function to be decreasing, we need to determine the conditions on and the corresponding values of the function. After solving, we find that must be between -1 and 1.
Thus, the correct answer is (3).

Step 1: Analyze the function.
We differentiate the function to find the intervals where the function is increasing. The derivative is positive in the intervals and .
Step 2: Conclusion.
Thus, the function is increasing in . Final Answer:

Step 1: Differentiate: f'(x)=(a²-3a+2)·(bounded trig term)+(a-1)
Step 2: For f(x) to be decreasing for all x, require a-1\ge0.
Step 3: Hence a\ge1.

Since is an increasing function in and it has a root in .
Only statement I is correct.

f'(x)=xˣ(ln x+1) f'(x)>0 when ln x+1>0⟹ x>\tfrac1e. This interval is not listed.

f'(x)=xˣ(ln x+1) f'(x)>0 when: ln x+1>0 ⟹ x>e⁻1 None of the given intervals exactly represent (e⁻1,∞).