Chapter 8

Condensing a Logarithmic Expression Condense the expression to the logarithm of a single quantity. $$3 \ln 4-\frac{1}{3} \ln \left(x^{2}+3\right)$$

Find values of \(x, y,\) and \(\lambda\) that satisfy the system. These systems arise in certain optimization problems in calculus. \((\lambda\) is called a Lagrange multiplier.) $$\left\\{\begin{aligned} y+\lambda &=0 \\ x &+\lambda=0 \\ x+y-10 &=0 \end{aligned}\right.$$

Condensing a Logarithmic Expression Condense the expression to the logarithm of a single quantity. $$2 \ln 7 t^{4}-\frac{3}{5} \ln t^{5}$$

Find values of \(x, y,\) and \(\lambda\) that satisfy the system. These systems arise in certain optimization problems in calculus. \((\lambda\) is called a Lagrange multiplier.) $$\left\\{\begin{aligned} 2 x &+\lambda=0 \\ 2 y+\lambda &=0 \\ x+y-4 &=0 \end{aligned}\right.$$

(a) determine the real zeros of \(f\) and (b) sketch the graph of \(f\). $$f(x)=x^{3}+x^{2}-12 x$$

(a) determine the real zeros of \(f\) and (b) sketch the graph of \(f\). $$f(x)=-8 x^{4}+32 x^{2}$$

(a) determine the real zeros of \(f\) and (b) sketch the graph of \(f\). $$f(x)=2 x^{3}+5 x^{2}-21 x-36$$

(a) determine the real zeros of \(f\) and (b) sketch the graph of \(f\). $$f(x)=6 x^{3}-29 x^{2}-6 x+5$$

Solve the equation. $$4 \sqrt{3} \tan \theta-3=1$$

Solve the equation. $$6 \cos x-2=1$$