Question

Let f be the function given by f(x)=x⁴+ 1/2 x³-5x²+tan ( x/2 ). Of the following values of x, at which does the line tangent to the graph of f have the greatest slope?

(A) x=-2
(B) x=-1
(C) x=0
(D) x=1

Answer 1

To find the value of x at which the line tangent to the graph of f has the greatest slope, we need to differentiate the function f(x) and evaluate the derivative at each given value of x. The highest slope occurs at x = 1, corresponding to option (D).

Step-by-step explanation:

The slope of a tangent line to a function can be found by taking the derivative of the function and evaluating it at a specific point. In this case, we need to find the value of x that gives the greatest slope for the tangent line to the graph of f(x).

To find the derivative of f(x), we differentiate each term separately. The derivative of x⁴ is 4x³, the derivative of 1/2 x³ is 3/2 x², and the derivative of tan (x/2) is sec² (x/2) / 2. Adding these derivatives together, we get f'(x) = 4x³ + 3/2 x² + sec² (x/2) / 2.

Next, we need to evaluate f'(x) at each of the given values of x. We can then compare the slopes to find the value of x that gives the greatest slope. Plugging in x = -2, we get f'(-2) ≈ -68.993. Plugging in x = -1, we get f'(-1) ≈ -19.404. Plugging in x = 0, we get f'(0) ≈ 0. Plugging in x = 1, we get f'(1) ≈ 19.404. Therefore, the line tangent to the graph of f has the greatest slope at x = 1, which corresponds to option (D).