Name: Inverse Function: Theorem
Uploaded: 2019-07-19
Duration: 4 min 27 s
Description: Did you know that only one-one and onto functions have an inverse? Play this video to learn more about the existence of inverse functions. Are you ready to e...

Question

Find the intervals of increase and decrease of the function

f (x) = a x^{2} + b x + c

, and its minimum and maximum values.

Accepted Answer

Let $f (x) = y = a x^{2} + b x + c \dots \dots (1)$

Now differentiate w.r.t $x$

$\Rightarrow \frac{d y}{d x} = 2 a x + b \dots \dots (2)$

Now the given function is increasing if $\frac{d y}{d x} > 0$

i.e., if $2 a x + b > 0$

Case- $(I)$ When $a > 0$

i.e., if $x > \frac{- b}{2 a}$

i.e., if $x \in (\frac{- b}{2 a}, \infty)$

Case- $(II)$ When $a < 0$

i.e. if $x < \frac{- b}{2 a}$

i.e. if $x \in (- \infty, \frac{- b}{2 a})$

And the given function is decreasing if $\frac{d y}{d x} < 0$

Again case- $(I)$ when $a > 0$

i.e., if $x < \frac{- b}{2 a}$

i.e., if $x \in (- \infty, \frac{- b}{2 a})$

Case- $(II)$ when $a < 0$

i.e. if $x > \frac{- b}{2 a}$

i.e. if $x \in (\frac{- b}{2 a}, \infty)$

And the function has minimum/maximum value if $\frac{d y}{d x} = 0$

i.e., if $x = \frac{- b}{2 a}$

Now, $\frac{d^{2} y}{d x^{2}} = 2 a$

Case- $(I)$ when $a > 0$

$\frac{d^{2} y}{d x^{2}} > 0$

$∴ f (x)$ has minimum value at $x = \frac{- b}{2 a}$

And minimum value of $f (x) = a {(\frac{- b}{2 a})}^{2} + b (\frac{- b}{2 a}) + c = \frac{4 a c - b^{2}}{4 a}$

$∴ f (x)$ has range $(\frac{4 a c - b^{2}}{4 a}, \infty)$ as $f (x)$ is increasing in $(- \frac{b}{2 a}, \infty)$ and $f (x) \to \infty when x \to \infty$ .

Case- $(II)$ When $a < 0$

$∴ \frac{d^{2} y}{d x^{2}} < 0$

$∴ f (x)$ has maximum value at $x = \frac{- b}{2 a}$ and maximum value is $\frac{4 a c - b^{2}}{4 a}$ .

In this case $f (x)$ has range $(- \infty, \frac{4 a c - b^{2}}{4 a})$ as $f (x)$ is decreasing in $(- \infty, \frac{- b}{2 a})$ .

Find the intervals of increase and decrease of the function $f (x) = a x^{2} + b x + c$ , and its minimum and maximum values.

Important Questions on Introduction to Mathematical Analysis

Learn and Prepare for any exam you want !

Find the intervals of increase and decrease of the function fx=ax2+bx+c, and its minimum and maximum values.

Important Questions on Introduction to Mathematical Analysis

Learn and Prepare for any exam you want !

Find the intervals of increase and decrease of the function $f (x) = a x^{2} + b x + c$ , and its minimum and maximum values.