Chapter 3: Problem 3
Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=x^{2}-6 x+8 $$
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Prove that if \(f^{\prime}(x)=0\) for all \(x\) in an interval \((a, b),\) then \(f\) is constant on \((a, b)\).
The function \(f\) is differentiable on the interval [-1,1] . The table shows the values of \(f^{\prime}\) for selected values of \(x\). Sketch the graph of \(f\), approximate the critical numbers, and identify the relative extrema. $$\begin{array}{|l|c|c|c|c|} \hline x & -1 & -0.75 & -0.50 & -0.25 \\ \hline f^{\prime}(x) & -10 & -3.2 & -0.5 & 0.8 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|} \hline \boldsymbol{x} & 0 & 0.25 & 0.50 & 0.75 & 1 \\ \hline \boldsymbol{f}^{\prime}(\boldsymbol{x}) & 5.6 & 3.6 & -0.2 & -6.7 & -20.1 \\ \hline \end{array}$$
The function \(s(t)\) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time \(t \geq 0\), (b) identify the time interval(s) when the particle is moving in a positive direction, (c) identify the time interval(s) when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction. $$ s(t)=t^{3}-20 t^{2}+128 t-280 $$
Find \(a, b, c,\) and \(d\) such that the cubic \(f(x)=a x^{3}+b x^{2}+c x+d\) satisfies the given conditions. Relative maximum: (2,4) Relative minimum: (4,2) Inflection point: (3,3)
The deflection \(D\) of a beam of length \(L\) is \(D=2 x^{4}-5 L x^{3}+3 L^{2} x^{2},\) where \(x\) is the distance from one end of the beam. Find the value of \(x\) that yields the maximum deflection.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
