Find the Critical Points f(x)=(2x^2+3)/(4x^2+5)

Problem

ƒ(x)=(2*x2+3)/(4*x2+5)

Solution

1. Identify the goal, which is to find the critical points by determining where the derivative ƒ(x)′ is equal to zero or undefined.

2. Apply the quotient rule for differentiation, which states that for a function u/v the derivative is (u′*v−u*v′)/(v2)

3. Calculate the derivatives of the numerator and denominator:

d(2*x2+3)/d(x)=4*x

d(4*x2+5)/d(x)=8*x

4. Substitute these into the quotient rule formula:

ƒ(x)′=((4*x)*(4*x2+5)−(2*x2+3)*(8*x))/((4*x2+5)2)

5. Simplify the numerator by distributing the terms:

ƒ(x)′=(16*x3+20*x−(16*x3+24*x))/((4*x2+5)2)

ƒ(x)′=(16*x3+20*x−16*x3−24*x)/((4*x2+5)2)

ƒ(x)′=(−4*x)/((4*x2+5)2)

6. Set the derivative to zero to find the critical points:

(−4*x)/((4*x2+5)2)=0

7. Solve for x by setting the numerator equal to zero:

−4*x=0

x=0

8. Check for points where the derivative is undefined by setting the denominator to zero:

(4*x2+5)2=0

4*x2+5=0

x2=−5/4

Since there are no real solutions for x2=−5/4 the derivative is defined for all real numbers.

Final Answer

x=0

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