c) Where is \( g(x)=0 \) ? (Enter a list of values, separated by commas.) d) Where is \( g^{\prime}(x)>0 \) ? (Enter an interval.) e) Where is \( g^{\prime}(x)<0 \) ? (Enter an interval.) t) Where is \( g^{\prime}(x)=0 \) ? (Enter a list of values, separated by commas.) (As needed above, use INF for \( \infty \), -INF for \( -\infty \), and U for the union symbol.)
Let \( f(x)=1-7 x^{2} \). If \( h \neq 0 \), then the difference quotient can be simplified as \[ \frac{f(x+h)-f(x)}{h}=A h+B x+C, \] where \( A, B \), and \( C \) are constants. (Note: It's possible for one or more of these constants to be 0 .) Find the constants. \[ A=, B= \] Use the simplified expression to find \( f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}= \) Finally, find each of the following: \[ f^{\prime}(1)=, f^{\prime}(2)= \]
Let \( f(x)=\frac{-4}{8 x+3} \). If \( h \neq 0 \), then the difference quotient can be simplified as \[ \frac{f(x+h)-f(x)}{h}=\frac{A}{(B x+C h+3)(D x+3)} \] where \( A, B, C \), and \( D \) are constants. (Note: It's possible for one or more of these constants to be 0 .) Find the bonstants. \[ \begin{array}{l} A=B= \\ \text { Use your answer from above to find } f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}= \end{array} \] Finally, find each of the following: \[ f^{\prime}(1)= \]