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Chapter 16: Problem 15

State the range of values of \(x\) for which each of the following functions is (i) concave up (ii) concave down: (a) \(y=\frac{x^{3}}{6}-\frac{5 x^{2}}{2}+3 x-9\) (b) \(y=3+x+\frac{x^{2}}{2}-\frac{x^{4}}{12}\) (c) \(y=\mathrm{e}^{x}-100 x-100\) (d) \(y=(x-1)^{4}\) (e) \(y=\frac{x^{2}}{2} \ln x-\frac{x^{2}}{4}\)

Short Answer

Expert verified
a) The function is concave up when x > 5 and concave down when x < 5. b) The function is concave up when x < -1 and x > 1, and concave down when -1 < x < 1.

Step by step solution

01

Find the second derivative

First, we need to find the first derivative of the function \(a\). The function is given by \(y(x)=\frac{x^{3}}{6}-\frac{5 x^{2}}{2}+3 x-9\). Differentiate it with respect to \(x\): \(y'(x) =\frac{1}{2}x^2 -5x+3\). Now, differentiate it again to find the second derivative: \(y''(x) = x-5\).
02

Find the critical points

Find the critical points by setting \(y''(x) = 0\), and solve for \(x\): \(x - 5 = 0\) \(x = 5\)
03

Test for concavity intervals

Test the intervals between critical points, that is \((x<5, x=5, x>5)\) on the second derivative \(y''(x)=x-5\): 1. \(x < 5\): \(y''(x) < 0\) - concave down 2. \(x = 5\): inflection point 3. \(x > 5\): \(y''(x) > 0\) - concave up
04

Write the answer for (a)

The function (a) is concave up when \(x > 5\) and concave down when \(x < 5\). #b)
05

Find the second derivative

The function (b) is given by \(y(x)=3+x+\frac{x^{2}}{2}-\frac{x^{4}}{12}\). Differentiate it with respect to \(x\) twice to find the second derivative: \(y'(x) =1 + x - \frac{x^3}{3}\) \(y''(x) =1-x^2\)
06

Find the critical points

Find the critical points by setting \(y''(x) = 0\), and solve for \(x\): \(1-x^{2}=0\) \(x=\pm 1\)
07

Test for concavity intervals

Test the intervals around the critical points, that is \((x<-1, x=-1, -11)\) on the second derivative \(y''(x)=1-x^{2}\): 1. \(x < -1\): \(y''(x) > 0\) - concave up 2. \(x = -1\): inflection point 3. \(-1 < x < 1\): \(y''(x) < 0\) - concave down 4. \(x = 1\): inflection point 5. \(x > 1\): \(y''(x) > 0\) - concave up
08

Write the answer for (b)

The function (b) is concave up when \(x<-1\) and \(x>1\), and concave down when \(-1 < x < 1\). Repeat the same steps for functions (c), (d), and (e).

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Most popular questions from this chapter

Find the equation of the normal to \(y=\ln x\) where \(x=2\).

Find the derivative of each of the following functions: (a) \(z(t)=\mathrm{e}^{3 t}(2 t-5)^{3}(3 t+1)^{4}\) (b) \(h(t)=3 \mathrm{e}^{-6 t} t^{7}(t+6)^{33}\) (c) \(M(p)=-p^{4} \sin ^{5} p\) (d) \(P(r)=6\left(1+r^{2}\right)^{6} \sqrt{3+r^{2}}\) (e) \(y(x)=\frac{(7-x)^{3}}{\left(3 x^{2}+1\right)^{2}}\)

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