Find the critical numbers of the function.
h(t) = t3/4 − 5t1/4
h = t^(3/4) - 5t^(1/4)
h' = 3/4 t^(-1/4) - 5/4 t^(-3/4)
= (3√t-5) / 4t^(3/4)
h' is undefined at t=0
h' = 0 at t = 25/9
To find the critical numbers of the function h(t) = t^3/4 − 5t^1/4, we need to find the values of t where the derivative of h(t) is equal to zero or undefined.
Let's start by finding the derivative of h(t). We can use the power rule to differentiate each term:
h'(t) = (3/4)t^(3/4 - 1) - (5/4)t^(1/4 - 1)
Simplifying the exponents:
h'(t) = (3/4)t^(-1/4) - (5/4)t^(-3/4)
To find the critical numbers, we set h'(t) equal to zero:
(3/4)t^(-1/4) - (5/4)t^(-3/4) = 0
Multiplying both sides by 4 to remove the fractions:
3t^(-1/4) - 5t^(-3/4) = 0
Now, let's solve for t.
First, let's move the term with t^(-3/4) to the other side:
3t^(-1/4) = 5t^(-3/4)
Next, divide both sides by t^(-3/4):
3t^(-1/4 - (-3/4)) = 5
Simplifying the exponents:
3t^0 = 5
Since any non-zero number raised to the power of 0 is equal to 1:
3(1) = 5
3 = 5
This equation is not true, so there are no solutions or critical numbers for this equation.
Therefore, the function h(t) = t^3/4 − 5t^1/4 has no critical numbers.
To find the critical numbers of a function, we need to find the values of t where the derivative of the function is either zero or undefined. The derivative of h(t) can be found using the power rule for differentiation.
Let's find the derivative of h(t):
h(t) = t^(3/4) - 5t^(1/4)
To find h'(t), we differentiate each term separately:
h'(t) = (3/4)t^(-1/4) - (5/4)t^(-3/4)
Now, to find the critical numbers, we need to set the derivative equal to zero and solve for t:
(3/4)t^(-1/4) - (5/4)t^(-3/4) = 0
We can notice that both terms have a common factor of (1/4)t^(-3/4). Factoring it out, we get:
(1/4)t^(-3/4)(3 - 5t) = 0
Now, we have two possibilities for the derivative to be zero:
1) (1/4)t^(-3/4) = 0
This equation has no solutions since t^(-3/4) cannot be zero.
2) 3 - 5t = 0
Solving this equation for t, we get:
5t = 3
t = 3/5
So, the only critical number of the function h(t) = t^(3/4) - 5t^(1/4) is t = 3/5.