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Question

# Let $f(x)=3^{3 x+1}-7 \cdot 5^{2 x}.$ a. Use the Maple commands solve and $f$ solve to try to ﬁnd all roots of $f.$ b. Plot $f(x)$ to ﬁnd initial approximations to roots of $f.$ c. Use Newton’s method to ﬁnd roots of $f$ to within $10^{-16}$ d. Find the exact solutions of $f(x)=0$ without using Maple.

Solution

Verified

Apply the function $\texttt{solve}$ in Maple as follows. The first argument is the equation we are trying to solve, the second one is the variable.

solve(3^(3*x+1)-7*5^(2*x)=0, x)


The result we get is

$-\dfrac{\ln{3}-\ln{7}}{3\ln{3}-2\ln{5}}$

Evaluate the expression to obtain the solution

$x=11.0094386$

The function fsolve is applied a bit differently, as shown below. The solution we get is the same as before.

f := 3^(3*x + 1) - 7*5^(2*x) = 0;
fsolve(f);
11.00943864


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