Practicing Success
Let $f$ be a differentiable function satisfying $f(x)+f(y)+f(z)+f(x) f(y) f(z)=14$ for all $x, y, z \in R$ Then, |
$f^{\prime}(x)<0$ for all $x \in R$ $f^{\prime}(x)=0$ for all $x \in R$ $f^{\prime}(x)>0$ for all $x \in R$ none of these |
$f^{\prime}(x)=0$ for all $x \in R$ |
We have, $f(x)+f(y)+f(z)+f(x) f(y) f(z)=14$ for all $x, y, z \in R$ ......(i) Putting x = y = z = 0, we get $3 f(0)+\{f(0)\}^3=14$ $\Rightarrow \{f(0)\}^3+3 f(0)-14=0$ $\Rightarrow f(0)=2$ Now, putting $y=z=x$ in (i), we get $3 f(x)+\{f(x)\}^3=14$ for all $x \in R$ Differentiating with respect to $x$, we get $3 f^{\prime}(x)+3\{f(x)\}^2 f^{\prime}(x)=0$ for all $x \in R$ $\Rightarrow \left\{\{f(x)\}^2+1\right\} f^{\prime}(x)=0$ for all $x \in R$ $\Rightarrow f^{\prime}(x)=0$ for all $x \in R$ |