extreme f(x)=e^x+4 — step-by-step solution

Find the strict local extrema of

e^{x} + 4

: they are the points of the domain where its derivative changes sign

⇥Let's study the variations of

f (x)

: first find where its derivative

f^{'} (x) = e^{x}

is zero

⇥Let's solve in

ℝ

e^{x} = 0

⇥In ℝ, the exponential function is strictly positive:

e^{x} > 0

, so

e^{x} = 0

has no solution

⇥Solution:

x \in \emptyset

⇥

f^{'} (x)

is never zero on the domain of

f

⇥Built from usual functions,

f^{'} (x)

is continuous where it is defined: on each interval where it is never zero, it keeps the sign of its value at any point

⇥

f^{'} (0) = 1

> 0:

f

is increasing on

ℝ

⇥Let's calculate the values of

f

at the bounds of the table, or its limits at the infinite or excluded bounds

⇥As

x

tends to

-∞

f (x)

tends to

4

⇥As

x

tends to

+∞

f (x)

tends to

+∞

⇥Let's draw up the variation table:

⇥

⇥So

f

is increasing on

ℝ

The derivative changes sign at no point of the domain: there is no strict local extremum

▶Strict local extrema:

\emptyset