Question

Statement I: The function $f(x)=x^2$ is decreasing in the interval $(0,\infty)$.
Statement II: Any function $y=f(x)$ is decreasing if $\dfrac{dy}{dx}& Lt;0$.
Which of the following is correct?

• A. Both the Statements I and II are true
• B. Both the Statements I and II are false
• C. Statement I is true and Statement II is false
• D. Statement I is false and Statement II is true

Hint:

If $\frac{dy}{dx}>0$, the function is increasing. If $\frac{dy}{dx}<0$, the function is decreasing.

Answer 1

The Correct Option is D

Step 1: Examine Statement I.
The given function is \[ f(x)=x^2 \] Compute its derivative \[ \frac{df}{dx}=2x \] For the interval \[ (0,\infty) \] we have \[ 2x>0 \] Thus the function is **increasing** in this interval, not decreasing. 
Therefore **Statement I is false**. 

Step 2: Examine Statement II. 
From calculus, if \[ \frac{dy}{dx}<0 \] then the slope of the function is negative. This means the function value decreases as \(x\) increases. 
Hence the function is **decreasing** in that interval. 
Thus **Statement II is true**. 

Step 3: Final conclusion. 
Statement I is false but Statement II is true. 
Final Answer: $\boxed{\text{Statement I is false and Statement II is true}}$