$13\mathrm{.}2$ Given

$f(x)=-1\mathrm{.}6x^6-3x^4+10x$

$($a$)$ Plot the function.

$($b$)$ Use analytical methods to prove that the function is concave for all values of $x.$

$($c$)$ Differentiate the function and then use a root$-$location method to solve for the maximum $f(x)$ and the corresponding value of $x.$

$13\mathrm{.}3$ Solve for the value of $x$ that maximizes $f(x)$ in Prob. $13\mathrm{.}2$ using the golden$-$section search. Employ initial guesses of $x_l=0$ and $x_u=1$ and perform three iterations.

$13\mathrm{.}4$ Repeat Prob. $13\mathrm{.}3,$ except use parabolic interpolation in the same fashion as Example $13\mathrm{.}2.$ Employ initial guesses of $x_0=0,x_1=1,$ and $x_2=2$ and perform three iterations.

$13\mathrm{.}5$ Repeat Prob. $13\mathrm{.}3$ but use Newton's method. Employ an initial guess of $x_0=1$ and perform three iterations.

$13\mathrm{.}6$ Employ the following methods to find the maximum of