Question

Write the composite function in the form f(g(x)).[Identify the inner function u = g(x)and the outer function

y = f(u).]
y = e⁷√ˣ
(g(x), f(u)) = ?? and find the derivative
For what values of x does the graph of fhave a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)
f(x) = x ? 2 sin x
x=??

Answer 1

The composite function f(g(x)) is y = e^(7th root of x). The derivative of f(g(x)) is (1/7)*(x^(-6/7))*e^(7th root of x). The graph of f(x) has a horizontal tangent at x = pi/3 + 2*pi*n, where n is an integer.

Step-by-step explanation:

To write the composite function in the form f(g(x)), we need to identify the inner function u = g(x) and the outer function y = f(u). In this case, the inner function is u = 7th root of x and the outer function is y = e^u. So, the composite function is y = e^(7th root of x).

To find the derivative of the composite function y = e^(7th root of x), we can use the chain rule. The derivative of e^u with respect to u is e^u, and the derivative of the 7th root of x with respect to x is (1/7)*(x^(-6/7)). Multiplying these derivatives together gives us the derivative of y with respect to x: dy/dx = (1/7)*(x^(-6/7))*e^(7th root of x).

The graph of f(x) has a horizontal tangent when the derivative of f(x) is equal to zero. In this case, f(x) = x - 2*sin(x). To find the values of x where the graph of f(x) has a horizontal tangent, we need to solve the equation 1 - 2*cos(x) = 0. Solving this equation gives us x = pi/3 + 2*pi*n, where n is an integer.