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

Problem

ƒ(x)=(2*x^2+3)/(4*x^2+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^′)/(v^2)

3. Calculate the derivatives of the numerator and denominator:

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

d(4*x^2+5)/d(x)=8*x

4. Substitute these into the quotient rule formula:

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

5. Simplify the numerator by distributing the terms:

ƒ(x)^′=(16*x^3+20*x−(16*x^3+24*x))/((4*x^2+5)^2)

ƒ(x)^′=(16*x^3+20*x−16*x^3−24*x)/((4*x^2+5)^2)

ƒ(x)^′=(−4*x)/((4*x^2+5)^2)

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

(−4*x)/((4*x^2+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*x^2+5)^2=0

4*x^2+5=0

x^2=−5/4

Since there are no real solutions for x^2=−5/4 the derivative is defined for all real numbers.

Final Answer

x=0

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