Question

Determine a cubic function with the given criteria.

Short answer

The resultant answer is \(f(x) = \frac{2}{9}{x^3} + \frac{1}{3}{x^2} - \frac{4}{3}x + \frac{7}{9}\).

Step-by-step solution

Given data

The given function is \(f(x) = \frac{2}{9}{x^3} + \frac{1}{3}{x^2} - \frac{4}{3}x + \frac{7}{9}\).

Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

Simplify the expression

Note that if a function \(f\) has a local maximum at \(a\), then we have\({f^\prime }(a) = 0\) and \({f^{\prime \prime }}(a) < 0\)

Analogously, if there is a local minimum at \(b\), then \({f^\prime }(b) = 0\) and \({f^{\prime \prime }}(b) > 0\).

Therefore, the given data implies the following:

$\(f( - 2) = 3,{f^\prime }( - 2) = 0,{f^{\prime \prime }}( - 2) < 0\)$

and

$$

f(1)=0, f^{\prime}(1)=0, f^{\prime \prime}(1)>0 .

$$

Simplify the expression using quadratic formula

The expression is: \({S^{\prime \prime }}(t) = 0\) only when \(\frac{{{p^2} - p}}{{{t^2}}} - \frac{{2pk}}{t} + {k^2} = 0\)

Using the Quadratic formula, we can write

\(\begin{aligned}{c}\frac{1}{t} &= \frac{{2pk \pm \sqrt {{{( - 2pk)}^2} - 4{k^2}\left( {{p^2} - p} \right)} }}{{2\left( {{p^2} - p} \right)}}\\\frac{1}{t} &= \frac{{2pk \pm \sqrt {4{p^2}{k^2} - 4{k^2}\left( {{p^2} - p} \right)} }}{{2\left( {{p^2} - p} \right)}}\\\frac{1}{t} &= \frac{{2pk \pm 2k\sqrt {{p^2} - \left( {{p^2} - p} \right)} }}{{2\left( {{p^2} - p} \right)}}\\\frac{1}{t} &= \frac{{pk \pm k\sqrt p }}{{\left( {{p^2} - p} \right)}}\end{aligned}\)

Take reciprocal of both sides:

\(\begin{aligned}{c}t &= \frac{{{p^2} - p}}{{pk \pm k\sqrt p }}\\t &= \frac{{{p^2} - p}}{{pk + k\sqrt p }}\quad \\t &= \frac{{{p^2} - p}}{{pk - k\sqrt p }}\end{aligned}\)

Substitute \(k = 0.07,p = 4\)

\(\begin{aligned}{c}t &= \frac{{{4^2} - 4}}{{4(0.07) + 0.07\sqrt 4 }}\\t &= 28.57\\t &= \frac{{{4^2} - 4}}{{4(0.07) - (0.07)\sqrt 4 }}\\t &= 85.71\end{aligned}\)

Sketch the graph

Graph for \(S(t)\) is shown below, the inflection points are marked by red dots.

At Inflection point the concavity of the graph reverses.