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Chapter 4: Problem 8

Sketch the graph of a function \(f\) that has a local minimum value at a point \(c\) where \(f^{\prime}(c)\) is undefined.

Short Answer

Expert verified
Answer: The primary characteristic of such a function is that it has a sharp turn, like a cusp or a corner, at the point \((c, f(c))\).

Step by step solution

01

Choose a point c for the local minimum

Choose a point on the coordinate plane where the function will have a local minimum value. Let this point be \((c, f(c))\).
02

Sketch a sharp turn at point c

At the point \((c, f(c))\), draw a cusp or corner, ensuring that the value of \(f(x)\) for \(x < c\) is higher than the value at \(c\) (i.e., the graph is decreasing), and that the value of \(f(x)\) for \(x > c\) is also higher than the value at \(c\) (i.e., the graph is increasing). This will create a sharp turn in the graph, causing the derivative \(f^{\prime}(c)\) to be undefined at the point \((c, f(c))\).
03

Connect the graph around point c

Sketch the remainder of the graph to the left and right of the point \((c, f(c))\), making sure that the graph shows a smooth transition from the decreasing region to the increasing region, maintaining the sharp turn at \((c, f(c))\). Since this is not a specific function, the shape of the graph before and after point \((c, f(c))\) is up to interpretation, as long as the overall conditions are met: the function has a local minimum at point \(c\), and its derivative is undefined at that point. By following the above steps, we have sketched a function \(f\) that has a local minimum value at point \(c\), and its derivative is undefined at that point.

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Most popular questions from this chapter

Suppose you make a deposit of \(S P\) into a savings account that earns interest at a rate of \(100 \mathrm{r} \%\) per year. a. Show that if interest is compounded once per year, then the balance after \(t\) years is \(B(t)=P(1+r)^{t}\) b. If interest is compounded \(m\) times per year, then the balance after \(t\) years is \(B(t)=P(1+r / m)^{m t} .\) For example, \(m=12\) corresponds to monthly compounding, and the interest rate for each month is \(r / 12 .\) In the limit \(m \rightarrow \infty,\) the compounding is said to be continuous. Show that with continuous compounding, the balance after \(t\) years is \(B(t)=P e^{n}\)

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