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Question

Mathematics Question on Maxima and Minima

Let
f(x)=3(x2−2)3+4,x∈R.\begin{array}{l} f\left(x\right)=3^{\left(x^2-2\right)^3+4},x\in \mathbb{R}.\end{array}
Then which of the following statements are true?
P : x = 0 is a point of local minima of f
Q : x = √2 is a point of inflection of f
R : f ′ is increasing for x > √2

A

Only P and Q

B

Only P and R

C

Only Q and R

D

All P, Q and R

Answer

All P, Q and R

Explanation

Solution

f(x)=3(x2−2)3+4,x∈R\begin{array}{l} f\left(x\right)=3^{\left(x^2-2\right)^3+4}, x\in R \end{array}
f(x)=81.3(x2−2)3\begin{array}{l} f\left(x\right)=81.3^{\left(x^2-2\right)^3} \end{array}
f’(x)=81.3(x2−2)3ln2.3(x2−2)2x\begin{array}{l} f’\left(x\right)=81.3^{\left(x^2-2\right)^3}\text{ln}2.3\left(x^2-2\right)2x\end{array}
=(486 ln2)(3(x2−2)3(x2−2)x)\begin{array}{l} =\left(486\ \text{ln}2\right)\left(3^{\left(x^2-2\right)^3}\left(x^2-2\right)x\right)\end{array}


⇒ x = 0 is the local minima.
f ′′(x) = (486 ln2)
(3(x2−2)3⋅(x2−2)(5x2−2+6x2ln3(x2−2)))\begin{array}{l} \begin{pmatrix}3^{\left(x^2-2\right)^3}\cdot\left(x^2-2\right) \\\\\left(5x^2-2+6x^2\text{ln}3\left(x^2-2\right)\right)\end{pmatrix}\end{array}
f”(x)=0     x=2\begin{array}{l} f”\left(x\right)=0~~~~~x=\sqrt{2}\end{array}
f”(2+)>0\begin{array}{l} f”\left(\sqrt{2}^+\right)>0\end{array}
⇒x=2\begin{array}{l} \Rightarrow x=\sqrt{2}\end{array}
is point of inflection
f”(x)>0 ∀ x>2\begin{array}{l} f”\left(x\right)>0~\forall~x>\sqrt{2}\end{array}
⇒ f ′(x) is increasing for x>2x > \sqrt2