h(x)= definite integral of ln(t^2)dt between 1 and x.

a) Evaluate h(1), h'(1), and h"(1).
b) Does h have a relative minimum, maximum, or neither at x = 1? Justify your answer.
c) Let k = h-1. Evaluate k'(0).

Question

h(x)= definite integral of ln(t^2)dt between 1 and x.

a) Evaluate h(1), h'(1), and h"(1).
b) Does h have a relative minimum, maximum, or neither at x = 1? Justify your answer.
c) Let k = h-1. Evaluate k'(0).

myininaya · Answer

$$h(x)=\int\limits_1^x \ln(t^2) dt $$
$$h(1)=\int\limits_1^1\ln(t^2) dt=?$$

anonymous · Answer

By the fundamental theorem of calculus,
$$\frac{d}{dx}\int_c^{g(x)}f(t)~dt=f(g(x))\cdot g'(x)$$
So,
$$h(x)=\int_1^x\ln t^2~dt~~\Rightarrow~~\begin{cases}h'(x)=\ln x^2\h''(x)=\dfrac{2}{x}\end{cases}$$
Part (b) is an application of the first derivative test. For part (c), you have $k=h-1$, which gives you
$$\begin{align*}k(x)&=(h-1)(x)\
&=h(x)-1\
&=\int_1^x\ln t^2~dt-1\
k'(x)&=\ln x^2
\end{align*}$$