Finding Critical Points

In E...

Question

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

f(x) = x(4 − x)³

Accepted Answer

Welcome back, everyone. Find all critical points and domain end points for the function H of C equals Z multiplied by 3 minus C, raise to the power of 4. Let's begin with the domain of this function. What we have to notice is that this is a factored form of a 5th degree polynomial, and we have to recall that polynomials are defined for all real numbers, meaning X belongs to the interval from negative infinity up to infinity. And this corresponds to our first answer to this problem. Now we're going to identify the critical points. We can find them by differentiating H of Z. We have to apply the product rule because we have a product of two terms. According to the product rule, we take the derivative of C. Multiply by 3 minus C erase the power of 4, and then we add C. Which is multiplied by the derivative of 3 minus C raises the power of 4. So we have the definition of the product rule, and then we can identify each derivative. The derivative of C is 1, we simply get 3 minus C erase the power of 4. Which is followed by the addition of C. Which is multiplied by the derivative of the composite functions we have to apply the chain rule. We begin with the exponents, so let's apply the power rule. We bring down the 4, which is multiplied by 3 minus C erase the power of 3. So we have applied the power rule and now according to the chain rule, we have to multiply by the derivative of 3 minus Z. Now the derivative of 3 minus C is -1. Because the derivative of 3 is 0, and the derivative of Z is 1, right? So we get -1. And we can simplify. So this gives us 3 minus C era to the power of 4. Plus 4 C multiplied by 3 minus C cubed multiplied by -1. So we can replace the positive sign with the negative sign. And now we can factor out 3 minus C cubed. Which leaves us with we minus C for the first term. And for the second term we are subtracting 4 C. So this gives us. 3 minus C cubed. Multiplied by 3 minus 5C. Now to identify the critical points we're going to set this derivative equals to 0. Which means that 3 minus C. Or 3 minus 5C is going to be equal to 0, right, because we have a factored form. So we can set each factor equal to 0. From the first part of our system of equations, we have C equals we. And from the second one, if we add 5 C to both sides and divide both sides by 5, we get C equals 3 divided by 5, so 3/5. Well, once we have our critical points. Z is a set of 2 numbers. 35 and 3. Let's label it as our second answer, and thank you for watching.

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

f(x) = x(4 − x)³

Key Concepts

Critical Points

Derivative

Domain Endpoints

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