let f be the logarithmic function given by f(x)=3log10x. which of the following statements about f is true…

Explanation:

Step1: Find the first - derivative

The derivative of $y = \log_{10}x$ is $y'=\frac{1}{x\ln 10}$. So for $f(x)=3\log_{10}x$, by the constant - multiple rule, $f'(x)=\frac{3}{x\ln 10}$. Since $x>0$ (domain of $\log_{10}x$) and $\frac{3}{\ln 10}>0$, $f'(x)>0$ for all $x > 0$. So $f(x)$ is increasing.

Step2: Find the second - derivative

Differentiate $f'(x)=\frac{3}{x\ln 10}=\frac{3}{\ln 10}x^{- 1}$ with respect to $x$. Using the power rule $(x^n)'=nx^{n - 1}$, we get $f''(x)=-\frac{3}{x^{2}\ln 10}<0$ for all $x>0$. A function is concave down when $f''(x)<0$.

Answer:

f is increasing, and the graph of f is concave down.