For every value of $x \in[1,3]$, the function $f(x)=\frac{1}{8^x}$ is

For every value of $x \in[1,3]$, the function $f(x)=\frac{1}{8^x}$ is
  1. increasing for $x>2$ and decreasing for $x \leq 2$.
  2. neither increasing nor decreasing.
  3. decreasing.
  4. increasing.

Solution

$\begin{aligned} & f(x)=\frac{1}{8^x}=8^{-x} \\ & \Rightarrow f^{\prime}(x)=8^{-x} \log 8 \times(-1) \\ & \Rightarrow f^{\prime}(x)=\frac{-\log 8}{8^x}<0 \forall x \in[1,3] \\ & \Rightarrow f(x) \text { is decreasing } \forall x \in[1,3]\end{aligned}$

Asked in: MHT CET 2022 (10 Aug Shift 2)

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