Final answer:
To find the local extrema of the function f(x) = x^1/5(x+24), we can find the critical points by taking the derivative of the function and setting it equal to zero. The critical points will indicate where the function may have local extrema.
Step-by-step explanation:
To find the local extrema of the function f(x) = x1/5(x+24), we need to find the critical points first. Critical points occur when the derivative of the function is equal to zero or undefined.
To find the derivative of the function, we can use the product rule. Let's denote g(x) = x1/5 and h(x) = (x+24).
g'(x) = (1/5)x-4/5
h'(x) = 1
Using the product rule, f'(x) = g(x)h'(x) + g'(x)h(x)
f'(x) = x1/5 + (1/5)x-4/5(x+24)
To find the critical points, we set f'(x) = 0 and solve for x:
x1/5 + (1/5)x-4/5(x+24) = 0
Solving this equation will give us the values of x where the function may have local extrema.