# Find the Local Maxima and Minima C(x)=375+0.75x^(3/4)

Find the first derivative of the function.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Combine and .
Multiply by .
Move to the denominator using the negative exponent rule .
Find the second derivative of the function.
Since is constant with respect to , the derivative of with respect to is .
Apply basic rules of exponents.
Rewrite as .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Combine and .
Move the negative in front of the fraction.
Differentiate using the Power Rule which states that is where .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Multiply and .
Multiply.
Multiply by .
Move to the denominator using the negative exponent rule .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Factor each term.
Factor out of .
Factor out of .
Separate fractions.
Divide by .
Combine and .
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Multiply each term by and simplify.
Multiply each term in by in order to remove all the denominators from the equation.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Since , there are no solutions.
No solution
Since there is no value of that makes the first derivative equal to , there are no local extrema.
No Local Extrema
Find the Local Maxima and Minima C(x)=375+0.75x^(3/4)