August 17th, 2014, 05:43 AM  #1 
Newbie Joined: Jul 2014 From: australia Posts: 16 Thanks: 0  graphs
Attached is the question i absolutely had so much trouble with this one. there is no answer in the back of the book for part 7 a. however, the answer for 7 b is: 3 1/8 units (this is a fraction, 3 and 1/8 units) 
August 17th, 2014, 08:22 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 
For part a) of the question, we need to equate the two functions and then solve for $x$: $\displaystyle x^23x=2xx^2$ Move everything to the left: $\displaystyle 2x^25x=0$ Factor: $\displaystyle x(2x5)=0$ Now, using the zerofactor property, we equate each factor in turn to zero and solve for $x$: $\displaystyle x=0$ $\displaystyle 2x5=0\implies x=\frac{5}{2}$ Thus, we have shown that the two graphs intersect for: $\displaystyle x\in\left\{0,\frac{5}{2}\right\}$ For part b), we may describe the vertical separation between the two graphs on the given interval as the difference between the upper graph and the lower graph: $\displaystyle d(x)=\left(2xx^2\right)\left(x^23x\right)=5x2x^2=x(52x)$ Now, this resulting quadratic function has two roots: $\displaystyle x\in\left\{0,\frac{5}{2}\right\}$ Because we see the squared term has a negative coefficient, we know the vertex will be its maximum. And we know the axis of symmetry will lie midway between the two roots. Thus the axis of symmetry is: $\displaystyle x=\frac{5}{4}$ And so the maximum vertical separation is: $\displaystyle d_{\max}= d\left(\frac{5}{4}\right)= \frac{5}{4}\left(5\frac{5}{2}\right)= \frac{5}{4}\cdot\frac{5}{2}= \frac{25}{8}$ Last edited by skipjack; August 17th, 2014 at 10:24 AM. 
August 17th, 2014, 10:42 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,036 Thanks: 2274 
Let f(x) = x²  3x and g(x) = 2x  x², so the graphs are of y = f(x) and y = g(x). As f(0) = g(0) = 0 and f(2$\small\frac12$) = g(2$\small\frac12$) = 1$\small\frac14$, the graphs meet where x = 0 and x = 2$\small\frac12$. The vertical separation is f(x)  g(x) = 2x²  5x, and the previous post shows how to proceed from there. 

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