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Removable Discontinuities

Let f x = ax ...

Question

Let $f (x) = a x^{2} + b x + c,$ where $a$ is positive and $b$ and $c$ are both negative, then the number of point in $R$ where $g (x) = f (| x |)$ is non-differentiable is/are

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Solution

$f (x) = a x^{2} + b x + c$
$y = f (x)$
f(|x|) = y $\to$

g(x) = |f(|x|)| $\to$

As shown in the figure there are 3 points at which $f (x)$ is non differentiable

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Similar questions

Q. Assertion :Let f : R

\to

R be defined as

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

, where a, b, c

ε

\neq

f (x) = 0

is having non-real roots, then

\int \frac{d x}{f (x)} = λ t a n^{- 1} (g (x)) + k

λ

k

are constants and

g (x)

is linear function of

x

t a n (t a n^{- 1} g (x)) = g (x) \forall x \in R

Q. Statement 1 : If

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

a > 0, c < 0

b \in R

, then roots of

f (x) = 0

must be real and distinct .
Statement 2 : If

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

a > 0, b \in R, b \neq 0

and the roots of

f (x) = 0

are real and distinct, then

c

is necessarily negative real number .

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

g (x) = A x^{2} + b x + λ,

a \neq 0, A \neq 0, a, b, c, A, λ \in R

f (x) = 0

g (x) = 0

are imaginary, then which of the following may be correct?

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

g (x) = A x^{2} + b x + λ,

a \neq 0, A \neq 0, a, b, c, A, λ \in R

f (x) = 0

g (x) = 0

are imaginary, then which of the following may be correct?

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

a, b, c

are rational, and

f : Z \to Z,

Z

is the set of integers. Then

a + b

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