Question

Let $f:\left[0,2\right]\to R$be a twice differentiable functions such that $f"\left(x\right)>0$ for all $x\in \left(0,2\right)$. If $\varphi \left(x\right)=f\left(x\right)+f\left(2-x\right).$ then $\varphi$ is:

• A. Increasing in $\left(0,1\right)$ and decreasing in $\left(1,2\right)$
• B. Decreasing in $\left(0,1\right)$ and increasing in $\left(1,2\right)$
• C. Increasing in $\left(0,2\right)$
• D. Decreasing in $\left(0,2\right)$

Answer 1

The correct option is B

Decreasing in $\left(0,1\right)$ and increasing in $\left(1,2\right)$

Explanation for correct option:

Step-1 Derivative of $\varphi \left(x\right)$

Consider the given equation as,

$\varphi \left(x\right)=f\left(x\right)+f\left(2-x\right)$

Differentiate the above Equation as,

$\varphi \text{'}\left(x\right)=f\text{'}\left(x\right)-f\text{'}\left(2-x\right)$

From the given data

$f"\left(x\right)>0$

$f\text{'}\left(x\right)$ is strictly increasing function

Step-2 : Increasing function condition:

For increasing function $\varphi \left(x\right)$

$$\begin{gathered} \Rightarrow \varphi \text{'}\left(x\right)>0 \\ \Rightarrow f\text{'}\left(x\right)-f\text{'}\left(2-x\right)>0 \\ \Rightarrow f\text{'}\left(x\right)>f\text{'}\left(2-x\right) \\ \Rightarrow x>2-x \\ \Rightarrow 2x>2 \\ \Rightarrow x>1 \end{gathered}$$

Thus, for the interval $\left[\left(1,2\right)\to f\left(x\right)\right]$ is increasing function. since $x$ belongs to $\left(1,2\right)$

Step-3 : Decreasing function condition:

For decreasing of function $\varphi \left(x\right)$

$$\begin{gathered} \Rightarrow \varphi \text{'}\left(x\right)<0 \\ \Rightarrow f\text{'}\left(x\right)-f\text{'}\left(2-x\right)<0 \\ \Rightarrow f\text{'}\left(x\right)<f\text{'}\left(2-x\right) \\ \Rightarrow x<2-x \\ \Rightarrow 2x<2 \\ \Rightarrow x<1 \end{gathered}$$

For interval $\left[\left(0,1\right)\to f\left(x\right)\right]$ is decreasing function. since $x$ belongs to $\left(0,1\right)$

Therefore, $\varphi$ is decreasing in $\left(0,1\right)$ and increasing in $\left(1,2\right)$

Hence, the correct answer is option (B).