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Analyzing Graph of Function : To analyze the graph of a given function we use the derivatives. We use various technique to analyze the behavior of the graph. Let us summarize the result of analyzing graph using the following table.
f(x) = $\frac{x^{2}4}{x^{2} 25}$
f(x) = $\frac{(x)^{2}4}{(x)^{2} 25}$
= $\frac{x^{2}4}{x^{2} 25}$
f(x) = f(x) so the function is even.
Step III : Find x and y intercepts
To find xintercept , plug in y=0 in the given function
f(x) = $\frac{x^{2}4}{x^{2} 25}$
0 = $\frac{x^{2}4}{x^{2} 25}$
$x^{2}$4 = 0
$x^{2}$ = 4
x = $\pm 2$
Xintercept : (2,0) and (2,0)
To find the yintercept, plug in x = 0 in the given function
f(x) = $\frac{x^{2}4}{x^{2} 25}$
f(0) = $\frac{0^{2}4}{0^{2} 25}$
y = $\frac{4}{25}$
y = 4/25
Yintercept : (0,4/25)
Step IV : Find vertical asymptotes
For vertical asymptotes, set the denominator equal to zero
$x^{2}$  25 = 0
$x^{2}$ = 25
x = $\pm$ 5
Vertical asymptote : x = 5 and x = 5
Step V : Find horizontal asymptotes
As the degree of the numerator is equal to the degree of denominator then the horizontal asymptote is
y = $\frac{Leading coefficient of numerator}{Leading coefficient of denominator} $
so Horizontal asymptote : y = 1
Step IV : Find the derivatives for( differentiability, Relative extrema, Concavity, Points of inflection)
f(x) = $\frac{x^{2}4}{x^{2} 25}$
=$\frac{(x^{2} 25)\frac{ \text{d}}{\text{d}x}(x^{2} 4)(x^{2} 4)\frac{\text{d}}{\text{d}x}(x^{2}25)}{(x^{2}25)^{2}}$
=$=\frac{(x^{2} 25)(2x)(x^{2} 4)(2x)}{(x^{2}25)^{2}}$
f '(x) = $\frac{42x}{(x^{2} 25)^{2}}$
f "(x) = $ \frac{42(3x^{2}+25) }{(x^{2} 25)^{3}}$
For critical points f '(x) = 0
$\frac{42x}{(x^{2} 25)^{2}}$ = 0
42x = 0
x= 0
f '(x) is undefined at x = $\pm$ 5
Interval for critical points : ($\infty$, 5) (5,0)(0,5) (5,$\infty$ )
Set the table :
(∞,5)  x=5 
(5,0) 
x=0 
(0,5) 
x=5 
(5,∞) 


f(x)  undf  4/25  undf  
f '(x)  +  undf  +  0    undf   
f "(x)  +  undf        undf  + 
Conclusion  increasing  Increasing  decreasing  decreasing  
concave up  concave down  concave down  concave up  
Rel.max 
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