Question

Find a function f that has the given derivative f' and value f(c), if possible.

$f\text{'}\left(x\right)=3-{\frac{4}{5}}^x,f\left(1\right)=0$

Short answer

The function is $3\mathrm{x}-\frac{{\frac{4}{5}}^{\mathrm{x}}}{\log {\frac{4}{5}}^{}}-2.992$.

The value of f(2.992( is$2.992$

Step-by-step solution

Step 1. Given

The given derivative of function is$f\text{'}\left(x\right)=3-{\frac{4}{5}}^x,f\left(1\right)=0$

Step 2.  Finding the function using integration 

$$\begin{gathered} f\text{'}\left(x\right)=3-{\frac{4}{5}}^x,f\left(1\right)=0 \\ \int f\text{'}\left(x\right)=\int 3-{\frac{4}{5}}^{x}dx \\ =3x-\frac{{\frac{4}{5}}^x}{\log {\frac{4}{5}}^{}}+c \\ \mathrm{where}\mathrm{c}\mathrm{is}\mathrm{constant} \\ \mathrm{f}\left(1\right)=3\left(1\right)-\frac{{\frac{4}{5}}^1}{\log {\frac{4}{5}}^{}}+\mathrm{c} \\ 0=3-\frac{4}{5}\times 0.0969+\mathrm{c} \\ 0=2.992+\mathrm{c} \\ \mathrm{c}=-2.992 \\ \mathrm{Hence}\mathrm{the}\mathrm{function}\mathrm{is}3\mathrm{x}-\frac{{\frac{4}{5}}^{\mathrm{x}}}{\log {\frac{4}{5}}^{}}-2.992 \end{gathered}$$

Step 3. Finding the value of f(c). 

$$\begin{gathered} f\left(x\right)=3\mathrm{x}-\frac{{\frac{4}{5}}^{\mathrm{x}}}{\log {\frac{4}{5}}^{}}-2.992 \\ Puttingvalueofc=2.992 \\ f\left(c\right)=3\times 2.992\frac{{\frac{4}{5}}^{2.992}}{\log {\frac{4}{5}}^{2.992}}-2.992 \\ f\left(c\right)=3.45 \end{gathered}$$