If f(x)=∫ limits0x et2(t-2)(t-3) d t for all x ∈(0, ∞), then

Question

If f(x)=∫ limits0x et2(t-2)(t-3) d t for all x ∈(0, ∞), then (A) f has a local maximum at x=2 (B) f is decreasing on (2,3) (C) there exists some c ∈(0, ∞) such that f prime(c)=0 (D) f has a local minimum at x=3

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/f4a86696bf36c25f0ebb8cd69c48bd11-.png" /> f'(x)=ex2(x-2)(x-3) Clearly, maxima at x=2, minima at x=3 and decreasing in x ∈(2,3). f'(x)=0 for x=2 and x=3 (Rolle's theorem) so there exist c ∈(2,3) for which f''(c)=0

Q. If $f (x) = 0 \int x e^{t^{2}} (t - 2) (t - 3) d t$ for all $x \in (0, \infty),$ then

$f$ has a local maximum at $x = 2$

$f$ is decreasing on (2,3)

there exists some $c \in (0, \infty)$ such that $f^{'} (c) = 0$

$f$ has a local minimum at $x = 3$

$f^{'} (x) = e^{x^{2}} (x - 2) (x - 3)$
Clearly, maxima at $x = 2,$ minima at $x = 3$ and
decreasing in $x \in (2, 3)$ .
$f^{'} (x) = 0$ for $x = 2$ and $x = 3$ (Rolle's theorem)
so there exist $c \in (2, 3)$ for which
$f^{''} (c) = 0$

Q. If f(x)=0∫x​et2(t−2)(t−3)dt for all x∈(0,∞), then

f has a local maximum at x=2

f is decreasing on (2,3)

there exists some c∈(0,∞) such that f′(c)=0

f has a local minimum at x=3