Question

Let f be a twice-differentiable function with f(0)=4. The derivative of f is given by f′(x)=sin(x2−2x+1) for −2≤x≤2.

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.

(a) Find all values of x in the interval −2 0 and f′′(x)<0. Is the approximation found in part (b) an overestimate or an underestimate for f(0. 25) ? Give a reason for your answer.

(d) Using the Mean Value Theorem, explain why the average rate of change of f over the interval −2≤x≤2 cannot equal 1. 25

Answer 1

For part (a), to find the values of x where f''(x)<0, we need to find the second derivative of f. The second derivative of f is given by the derivative of f', which is

f''(x) = cos(x^2 - 2x + 1) * (-2x + 2).

For f''(x)<0, cos(x^2 - 2x + 1) must be positive and (-2x + 2) must be negative. The cosine function is positive for all values of x, so we need to find the values of x such that -2x+2 is negative. This inequality simplifies to x>1. Therefore, the values of x in the interval -2≤x≤2 where f''(x)<0 are 1
For part (b), we can use the second derivative test to determine whether the approximation is an overestimate or underestimate. If f''(x)>0 at x=0.25, then the approximation is an overestimate. If f''(x)<0 at x=0.25, then the approximation is an underestimate. Since we found that f''(x)<0 for x>1, and 0.25 is less than 1, the approximation is an underestimate for f(0.25).

For part (d), the Mean Value Theorem states that for a function f defined on an interval [a,b], there exists a value c in the interval (a,b) such that f'(c) is equal to the average rate of change of f over the interval [a,b]. In this case, the average rate of change of f over the interval -2≤x≤2 is equal to (f(2) - f(-2))/(2-(-2)) = (f(2) - f(-2))/4. Since f is a function and its value can be any real number, the average rate of change of f over the interval -2≤x≤2 can be any real number. Therefore, it is not possible for the average rate of change of f over the interval -2≤x≤2 to equal 1.25.

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