Let f be a twice-differentiable function defined on the interval -1.2 less than or equal to x less than or equal to 3.2 with f(1)=2. The graph of f', the derivative of f, is shown on the right. The graph of f' crosses the x-axis at x=-1 and x=3 and has a horizontal tangent at x=2. Let g be the function given by g(x)=e^(f(x)). 1. Write an equation for the line tangent to the graph of g at x=1. 2. For -1.2 is less than or equal to x is less than or equal to 3.2, find all values of x at which g has a local maximum. Justify your answer. 3. The second derivative of g is g''(x)=x^(f(x)) [(f'(x))^2 + f''(x)]. Is g''(-1) positive, negative, or zero? Justify your answer. 4. Find the average rate of change of g', the derivative of g, over the interval [1,3].
To answer these questions, we need to follow a step-by-step approach. Let's break down each question and explain how to find the answer.
1. Write an equation for the line tangent to the graph of g at x=1:
To find the equation for the tangent line at x=1, we first need to find the value of g'(1), which represents the slope of the tangent line. Recall that g(x) = e^(f(x)). The derivative of g(x) with respect to x, denoted as g'(x), can be found by applying the chain rule: g'(x) = f'(x) * e^(f(x)). Since we are looking for g'(1), we need to find f'(1) and substitute it into the equation g'(x) = f'(x) * e^(f(x)).
2. Find all values of x at which g has a local maximum:
To find the local maximum of g, we need to locate the critical points where g'(x) = 0 or g'(x) is undefined. Since we know that g'(x) = f'(x) * e^(f(x)), we need to find the values of x where f'(x) = 0 or f'(x) is undefined. From the given information, we know that f' crosses the x-axis at x=-1 and x=3, which are the points where f'(x) = 0. So, those points are potential candidates for local maxima.
3. Determine if g''(-1) is positive, negative, or zero:
To find g''(x), we are given the formula g''(x) = x^(f(x)) * [(f'(x))^2 + f''(x)]. We need to evaluate g''(-1), so we substitute x = -1 into this formula and calculate the value. Depending on whether this value is positive, negative, or zero, we can determine the sign of g''(-1).
4. Find the average rate of change of g' over the interval [1,3]:
To find the average rate of change of g' over the interval [1,3], we need to calculate the difference between g'(3) and g'(1), and then divide it by the difference between 3 and 1 (which is the width of the interval [1,3]). The average rate of change of g' can be interpreted as the slope of the secant line between the points (1, g'(1)) and (3, g'(3)).
Please note that these explanations outline the general approach to answering the questions. To get the specific numeric answers, you would need to use the given information and make calculations accordingly.