Suppose that f has a negative derivative for all values of x...

Suppose that $f$ has a negative derivative for all values of $x$ and that $f(1)=0 .$ Which of the following statements must be true of the function $$ h(x)=\int_{0}^{x} f(t) d t ? $$ Give reasons for your answers. (a) $h$ is a twice-differentiable function of $x$. (b) $h$ and $d h / d x$ are both continuous. (c) The graph of $h$ has a horizontal tangent at $x=1$. (d) $h$ has a local maximum at $x=1$. (e) $h$ has a local minimum at $x=1$ (f) The graph of $h$ has an inflection point at $x=1$. (g) The graph of $d h / d x$ crosses the $x$ -axis at $x=1$.

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00:01 Here, we're given just a little bit of information about f and h.

00:06 So first off, f of x has a negative derivative for all values.

00:15 Also, we're told that f of 1 is equal to 0.

00:20 And then we're told that h of x is the integral from 0 to x of f of tdt.

00:29 So what i want to know now is, first off, a, is h a twice differentiable function? i could calculate h prime of x, that's just f of x.

00:43 And if i take its second derivative, that would be f prime of x.

00:49 So this really is asking, is f prime of x always exist? and yes, because it's less than zero.

00:58 So i'm assuming that being less than zero means that it's always defined.

01:04 If i look at option b, we're going to say h and dhdx are both continuous.

01:11 Well, if f is continuous, it's also, or sorry, if it's differentiable, it must also be continuous.

01:19 Also, when i take the integral of h, that will make it continuous.

01:23 So again, the answer is yes.

01:26 This has to be true as well.

01:29 Now let's look at option c says the graph of h has a horizontal tangent at x equals 1.

01:36 Well, h prime of x is going to be f of x, as we mentioned earlier.

01:44 If we were to set that equal to zero to find a critical point, you'd find that x equals 1 is a solution because we're told f of 1 is 0.

01:53 So the answer again is yes.

01:57 It has a horizontal tangent line because its derivative is zero.

02:02 For part d, does h have a local maximum? well, what we'd have to do next is take the second derivative test, right? if i take h prime of x is f of x, and then take the second derivative.

02:16 Derivative test, h double prime is f prime of x...

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