Question
This is the graph of $f′(x)$.
A function $f$ is defined on the interval $[0,4]$ with $f(2)=3, f(x)=e^{\sin x}-2 \cos (3 x)$, and $f^{\prime}(x)=(\cos x) \cdot e^{\sin x}+6$ $\sin (3 x)$
(a) Find all values of $x$ in the interval where $f$ has a critical point. Classify each critical point as a local maximum, local minimum, or neither. Justify your answers.
(b) On what subinterval(s) of $(0,4)$, if any, is the graph of $f$ concave down? Justify your answer.
(c) Write an equation for the line tangent to the graph of $f$ at $x=1.5$.
▶️Answer/Explanation
(a) The graph of $f$ has critical points when $f(x)=0$. This occurs at points $a$ and $b$ on the graph of $f(x)$ above, where $a=0.283$ and $b=3.760$.
At $x=0.283$, the graph of $f$ has a local minimum because $f(x)$ changes from negative to positive at that value.
At $x=3.760$, the graph of $f$ has a local maximum because $f(x)$ changes from positive to negative at that value.
(b) $f^{\prime}(x)=0$ at points $p, q$, and $r$ on the graph of $f(x)$ because that is where the graph of $f(x)$ has horizontal tangent lines, where $p=1.108, q=2.166$, and $r=3.082$. Note, you could also graph $f^{\prime \prime}(x)$ and find the roots of the graph.
To determine the intervals where $f$ is concave down, look for decreasing intervals on $f^{\prime \prime}(x)$ or look for intervals on the graph of $f^{\prime \prime}(x)$ where $f^{\prime \prime}(x)<0$.
The intervals where $f$ is concave down are $(p, q)=(1.108,2.166)$ and $(r, 4)=(3.082,4)$.
(c) $f(1.5)=3.1330726$
$
f(1.5)=f(2)+\int_2^{1.5} f^{\prime}(x) d x=3+\int_2^{1.5} f^{\prime}(x) d x=2.1409738
$
An equation of the tangent line at $x=1.5$ is $y=2.141+3.133(x-1.5)$