Question

Let f be the function defined by f(x)=2x+e ˣ. If g(x)=f⁻¹ (x) for all x and the point (0,1) is on the graph of f, what is the value of g′(1) ?

(A) 1 / 2+e
(B) 1/3
(C) 1/ 2
(D) 3
(E) 2+e

Answer 1

The derivative of the inverse function, g', at the point 1 is the reciprocal of the derivative of the original function f at its corresponding point. Since f'(0) is 3, g'(1) equals 1/3, which is answer (B).

Step-by-step explanation:

To find the value of g′(1), where g is the inverse function of f, we need to use the fact that given a function f and its inverse g, the derivatives of these functions at corresponding points are reciprocals of each other. That is,

f′(a) = 1 / g′(b) if f(a) = b and g(b) = a.

Since f(0) = 2(0) + e^0 = 1, we know that g(1) = 0. To find f′(x), we differentiate f(x) = 2x + e^x to get f′(x) = 2 + e^x. Therefore, f′(0) = 2 + e^0 = 2 + 1 = 3.

Now, using the reciprocal relationship, g′(1) = 1 / f′(0) = 1 / 3.

The correct answer is (B) 1/3.