6.5: Exercises

6.1. Zeros, local minima and maxima. A zero of a function is a place where \(f(x)=0\).

(a) Find the zeros, local maxima, and minima of the polynomial \(y=\) \(f(x)=x^{3}-3 x\)

(b) Find the local minima and maxima of the polynomial \(y=f(x)=\) \((2 / 3) x^{3}-3 x^{2}+4 x\)

(c) Determine whether each of the polynomials given in parts (a) and (b) have an inflection point.

6.2. Classifying critical points. Find critical points, zeros, and inflection points of the function \(y=f(x)=x^{3}-a x\). Then classify the types of critical points that you have found.

6.3. Sketching graphs. For each of the following functions, sketch the graph for \(-1<x<1\), find \(f^{\prime}(0), f^{\prime}(1), f^{\prime}(-1)\) and identify any local minima and maxima.

(a) \(y=x^{2}\)

(b) \(y=-x^{3}\)

(c) \(y=-x^{4}\)

(d) Using your observations, when can you conclude that a function whose derivative is zero at some point has a local maximum at that point?

6.4. Sketching a graph. Sketch a graph of the function \(y=f(x)=x^{4}-2 x^{3}\), using both calculus and methods of Chapter 1 .

6.5. Global maxima and minima. Find the global maxima and minima for the function in Exercise 4 on the interval \(0 \leq x \leq 3\).

6.6. Absolute maximum and minimum. Find the absolute maximum and minimum values on the given interval:

(a) \(y=2 x^{2}\) on \(-3 \leq x \leq 3\)

(b) \(y=(x-5)^{2}\) on \(0 \leq x \leq 6\)

(c) \(y=x^{2}-x-6\) on \(1 \leq x \leq 3\)

(d) \(y=\frac{1}{x}+x\) on \(-4 \leq x \leq-\frac{1}{2}\).

6.7. Local vs. absolute. A function \(f(x)\) has as its derivative \(f^{\prime}(x)=\) \(2 x^{2}-3 x\)

(a) In what regions is \(f\) increasing or decreasing?

(b) Find any local maxima or minima.

(c) Is there an absolute maximum or minimum value for this function?

6.8. Minimum value. Sketch the graph of \(x^{4}-x^{2}+1\) in the range \(-3\) to 3 . Find its minimum value.

6.9. Critical points. Identify all the critical points of the following function.

\[y=x^{3}-27 \nonumber \]

6.10. Critical and inflection points. Consider the function \(g(x)=x^{4}-\) \(2 x^{3}+x^{2}\). Determine locations of critical points and inflection points.

6.11. No critical points. Consider the polynomial \(y=x^{3}+3 x^{2}+a x+1\). Show that when \(a>3\) this polynomial has no critical points.

6.12. Critical points and generic parabola. Find the values of \(a, b\), and \(c\) if the parabola \(y=a x^{2}+b x+c\) is tangent to the line \(y=-2 x+3\) at \((2,-1)\) and has a critical point when \(x=3\).

6.13. Double wells and physics. In physics, a function such as

\[f(x)=x^{4}-2 x^{2} \nonumber \]

is often called a double well potential. Physicists like to think of this as a "landscape" with hills and valleys. They imagine a ball rolling along such a landscape: with friction, the ball eventually comes to rest at the bottom of one of the valleys in this potential. Sketch a picture of this landscape and use information about the derivative of this function to predict where the ball might be found, i.e. where the valley bottoms are located.

6.14. Function concavity. Find the first and second derivatives of the function

\[y=f(x)=\frac{x^{3}}{1-x^{2}} \nonumber \]

Use information about the derivatives to determine any local maxima and minima, regions where the curve is concave up or down, and any inflection points.

6.15. Classifying critical points. Find all the critical points of the function

\[y=f(x)=2 x^{3}+3 a x^{2}-12 a^{2} x+1 \nonumber \]

and determine what kind of critical point each one is. Your answer should be given in terms of the constant \(a\), and you may assume that \(a>0\).

6.16. Describing a function. The function \(f(x)\) is given by \[y=f(x)=x^{5}-10 k x^{4}+25 k^{2} x^{3} \nonumber \] where \(k\) is a positive constant.

(a) Find all the intervals on which \(f\) is either increasing or decreasing. Determine all local maxima and minima.

(b) Determine intervals on which the graph is either concave up or concave down. What are the inflection points of \(f(x)\) ?

6.17. Muscle shortening. In 1938 Av Hill proposed a mathematical model for the rate of shortening of a muscle, \(v\), (in \(\mathrm{cm} / \mathrm{sec}\) ) when it is working against a load \(p\) (in gms). His so called force-velocity curve is given by the relationship

\[(p+a) v=b\left(p_{0}-p\right) \nonumber \]

where \(a, b, p_{0}\) are positive constants.

(a) Sketch the shortening velocity versus the load, i.e., \(v\) as a function of \(p\).

Note: the best way to do this is to find the intercepts of the two axes, i.e. find the value of \(v\) corresponding to \(p=0\) and vice versa.

(b) Find the rate of change of the shortening velocity with respect to the load, i.e. calculate \(d v / d p\).

(c) What is the largest load for which the muscle contracts? (hint: a contracting muscle has a positive shortening velocity, whereas a muscle with a very heavy load stretches, rather than contracts, i.e. has a negative value of \(v\) ).

6.18. Reaction kinetics. Chemists often describe the rate of a saturating chemical reaction using Michaelis-Menten \(\left(R_{m}\right)\) or sigmoidal \(\left(R_{s}\right)\) kinetics

\[R_{m}(c)=\frac{K c}{k_{n}+c}, \quad R_{S}(c)=\frac{K c^{2}}{k_{n}^{2}+c^{2}} \nonumber \]

where \(c\) is the concentration of the reactant, \(K>0, k_{n}>0\) are constants. \(R(c)\) is the speed of the reaction as a function of the concentration of reactant.

(a) Sketch the two curves. To do this, you should analyze the behavior for \(c=0\), for small \(c\), and for very large \(c\). You will find a horizontal asymptote in both cases. We refer to that asymptote as the "maximal reaction speed". What is the "maximal reaction speed" for each of the functions \(R_{m}, R_{s}\) ?

Note: express your answer in terms of the constants \(K, k_{n}\).

(b) Show that the value \(c=k_{n}\) leads to a half-maximal reaction speed. For the questions below, you may assume that \(K=1\) and \(k_{n}=1\).

(c) Show that sigmoidal kinetics, but not Michaelis-Menten kinetics has an inflection point.

(d) Explain how these curves would change if \(K\) is increased; if \(k_{n}\) is increased.

6.19. Checking the endpoints. Find the absolute maximum and minimum values of the function \[f(x)=x^{2}+\frac{1}{x^{2}} \nonumber \] on the interval \(\left[\frac{1}{2}, 2\right]\). Be sure to verify if any critical points are maxima or minima and to check the endpoints of the interval.