Home / AP Calculus AB : 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema- Exam Style questions with Answer- FRQ

AP Calculus AB : 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema- Exam Style questions with Answer- FRQ

Question:

The continuous function f is defined on the interval -4 ≤  x ≤ 3. The graph of f consists of two quarter circles and one line segment, as shown in the figure above. Let \(g(x)=2x+\int_0^x f(t)dt\).

(a) Find g(-3) . Find g'(x) and evaluate g'(-3) .
(b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval -4 ≤  x ≤ 3. Justify your answer.
(c) Find all values of x on the interval -4 ≤  x ≤ 3 for which the graph of g has a point of inflection. Give a reason for your answer.
(d) Find the average rate of change of f on the interval-4 ≤  x ≤ 3.  There is no point c, -4 < c < 3,  for which f'(c) is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.

▶️Answer/Explanation

Ans:

(a)

\(g(-3)=2(-3)+\int_{0}^{-3}f(t)dt = -6 -\frac{9\pi }{4}\)

\(g'(x)=\frac{d}{dx}\left ( 2x + \int_{0}^{x}f(t)dt \right )=2+f(x)\)

g'(-3) = 2 + f(-3) = 2 + 0 = 2

(b)

g'(x) = 0                        2 + f(x) = 0 

g'(x)                              f(x) = -2

                                       x = 5/2

                                    \(g(-4)=-8+\int_{0}^{-4}f(t)dt\)

                                    = -8 – 2π

\(g(5/2)= 5+\int^{\frac{5}{2}}_{0} f(t)dt = + \frac{5}{4}\)

\(g(3)= 6+\int_{0}^{3}f(t)dt = 6\)

x = 5/2 , because g’   going from t to – proves it as the only relative maximum and g(5/2) is greater than g at either endpoint.   

(c)

f'(x)                          g”(x) = d/dx    (g(x)) = f'(x)

The only point of inflection for g is at x = 0, since f'(x), which is equivalent to g”, only changes signs at x = 0 on the interval -4 ≤ x ≤ 3

(d)

Avg. Rate of change  = \(\frac{f(3)-f(-4)}{3-4}\)

\(\frac{-3-1}{3+4}=\frac{-2}{7}\)

Because mean value Theorem only applies when the function is continuous And differentiable on the interval, which doesn’t apply here since f(x) isn’t differentiable at x = 0.

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