Question

A. Tell whether each given function has a solution on the indicated closed interval. Prove using the Intermediate value theorem.

1. f(x)=3x²+2x²;[-1,1]

2. f(x)= 2-x²/x²;[-3,-1]

B. Sketch the graph of the following functions and then find the absolute extreme values of each of the given interval.

1.f(x)=√x²-25;[5,10]

2. f(x)=-1/x²;[0.5,2]

C. A restaurant's profit function (in hundreds) for hamburgers is given by the function P such that P(x)=1.22x-x²/30,000-4,000,where 0≤x≤20,000.

1. How many hamburgers does the restaurant need to sell to yield the maximum profit?

2. What is the maximum profit from the sale of hamburger ?

Answer 1

Solution (A)

Given that

$f(x)=3x^3+2x$ with in interval $[-1,1]$

$f(-1)=3(-1)^3+2(-1)=-5<0$

$f(1)=3(1)^3+2(1)=5>0$

Since

$f(-1)<0$ and $f(1)>0$

Therefore, according to IVT the function $f(x)=3x^3+2x$ has a solution within the interval $[-1,1]$

The plot above shows that the function $f(x)=3x^3+2x$ has a solution within the interval $[-1,1]$, which is $x=0$

Solution (B)

for the function

$g(x)=\sqrt{x^2-25}$ with in the interval $[5, 10]$ , the plot is shown below

From the plot, the extreme values are

The maximum value is $8.66$ at $x=10$

The minimum value is at $x=5$

Now For the function

$h(x)=-\frac{1}{x^2}$ within the interval $[0.5,2]$

The plot is shown below

From the plot, the extreme values are

The maximum value is $-0.25$ at $x=2$

The minimum value is $-4$ at $x=0.5$

Solution (C)

$p(x)=1.22x-\frac{x^2}{30000}-4000$ with in the interval $[0, 20000]$ , the plot is showb below

From the plot,

The maximum profit is 7163

And the hamburger sold were 18300