Question
(a) Topic-6.7- The Fundamental Theorem of Calculus and Definite Integrals
(b) Topic-5.2- Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
(c) Topic-5.3- Determining Intervals on Which a Function is Increasing or Decreasing
(d) Topic-5.5- Using the Candidates Test to Determine Absolute (Global) Extrema
3. Let f be a differentiable function with f(4) = 3. On the interval \(0\leq x\leq 7\), the graph of f ‘, the derivative of f , consists of a semicircle and two line segments, as shown in the figure above.
(a) Find f( 0) and f(5).
(b) Find the x-coordinates of all points of inflection of the graph of f for \(0\leq x\leq 7\). Justify your answer.
(c) Let g be the function defined by g( x) = f(x ) − x. On what intervals, if any, is g decreasing for \(0\leq x\leq 7\) ? Show the analysis that leads to your answer.
(d) For the function g defined in part (c), find the absolute minimum value on the interval \(0\leq x\leq 7\). Justify your answer.
▶️Answer/Explanation
3(a) Find f (0 ) and f (5).
\(f(o)=f(4)+\int_{4}^{0}f'(x)dx=3-\int_{0}^{4}f'(x)dx=3+2\pi\)
\(f(5)=f(4)+\int_{4}^{5}f'(x)dx= 3+\frac{1}{2}=\frac{7}{2}\)
3(b) Find the x -coordinates of all points of inflection of the graph of f for 0 < x < 7. Justify your answer.The graph of f has a point of inflection at each of x = 2 and x = 6, because f ′( x) changes from decreasing to increasing at x = 2 and from increasing to decreasing at x = 6.
3(c) Let g be the function defined by g( x) = f ( x) − x. On what intervals, if any, is g decreasing for 0 ≤ x ≤ 7 ? Show the analysis that leads to your answer.
g′(x ) = f ′( x) − 1
f ′( x) − ≤ 10 ⇒ f ′( x) ≤ 1
The graph of g is decreasing on the interval 0 ≤ x ≤ 5 because g′(x ) ≤ 0 on this interval.
3(d) For the function g defined in part (c), find the absolute minimum value on the interval 0 ≤ x ≤ 7. Justify your answer.
g is continuous, g′(x ) < 0 for 0 < x < 5, and g′(x ) > 0 for 5 < x < 7.
Therefore, the absolute minimum occurs at x = 5, and \(g(5)=f(5)-5=\frac{7}{2}-5=-\frac{3}{2}\) is the absolute minimum value of g.