Question 5
(a)-Topic-5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
(b)-Topic-5.6 Determining Concavity of Functions over Their Domains
(c)-Topic-5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
(d)-Topic-5.7 Using the Second Derivative Test to Determine Extrema
5. The functions f and g are twice differentiable. The table shown gives values of the functions and their first derivatives at selected values of x.
(a) Let h be the function defined by h(x) = f(g(x)). Find h'(7). Show the work that leads to your answer.
(b) Let k be a differentiable function such that k'(x) = \(\left ( f(x) \right )^{2}\) · g(x). Is the graph of k concave up or concave down at the point where x = 4 ? Give a reason for your answer.
(c) Let m be the function defined by m(x) = \(5x^{3}+\int_{0}^{x}f'(t)dt \). Find m(2). Show the work that leads to your answer.
(d) Is the function m defined in part (c) increasing, decreasing, or neither at x = 2 ? Justify your answer.
▶️Answer/Explanation
5(a) Let h be the function defined by h(x) = f(g(x)). Find h′(7 ). Show the work that leads to your answer.
\(h'(x) = f'(g(x)). g'(x)\)
\(h'(7) = f'(g(7)).g'(7)\)
\(= f'(0).8 = \frac{3}{2}.8 = 12\)
5(b) Let k be a differentiable function such that \(k'(x) = (f(x))^{2} .g(x)\). Is the graph of k concave up or concave down at the point where x = 4 ? Give a reason for your answer.
\(k”(x) = 2f(x) .f'(x) .g(x) + (f(x))^{2} .g'(x)\)
\(k”(4) = 2f(4) .f'(4) .g(4) + (f(4))^{2} .g'(4)\)
\(= 2.4.3.(-3) + 4^{2}.2 = -72 +32 = -40\)
The graph of k is concave down at the point where x = 4 because k′′(4 0 ) < and k′′ is continuous.
5(c) Let m be the function defined by \(m(x) = 5x^{3} + \int_{0}^{x} f'(t) dt\). Show the work that leads to your answer.
\(m(2) = 5.8 +\int_{0}^{2} f'(t) dt = 4= + (f(2) – f(0))\)
\(= 40 + (7-10) = 37\)
5(d) Is the function m defined in part (c) increasing, decreasing, or neither at x = 2 ? Justify your answer.
\(m'(x) = 15x^{2} + f'(x)\)
\(m'(2) = 15.4 + f'(2)= 60 +(-8) = 52\)