Question

If $g(t)=\,\lbrace max(t^3-6t^2+9t-3,0\,\,t \, belongs \,to [0,3])\rbrace\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lbrace\,4-t\,\,\,\,\,\,t\,belongs\,to\,(3,4)\rbrace$
Then the number of points at which $g(t)$ is non differentiable is:
(A) $1$
(B) $3$
(C) $2$
(D) $4$

Answer 1

Hint: In this question we have to find the number of points which are non-differentiable in the given equation $g(t)$ . Only we have to do is that we simply plot a graph and we know that where the function is discontinuous the points are our non-differentiable points. By which we can easily get the number of points.

Formula Used:
Common differential formula of power used, $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$

Complete step by step Solution:
Firstly, we take all the data which is provided in the question, after that we will start our solution,
We had received an equation let us assume it as,
$y = ({t^3} - 6{t^2} + 9t - 3)$
By taking differentiation of the above equation with respect to x, we get that,
$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}({t^3} - 6{t^2} + 9t - 3)$
By using differential identity in the above equation,
Identity used $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$
As per doing further solutions we get,
$\dfrac{{dy}}{{dx}} = 3{t^2} - 12t + 9$
As we take differentiable as $y^{'}$ and taking common in the above equation,
$y^{'} = 3({t^2} - 4t + 3)$
By doing factorize in the above equation we get the values of points which we have to plot on a graph,
As per doing a further calculation of factorization,
$y^{'} = 3(t - 1)(t - 3)$
By putting t’s value in 2nd equation given in the question, we get the values for plotting the graph,
As after which get all data, we have plotted the graph which is given below,


As per we get detail from the graph that there are $3$ points where the function is discontinuous.
Therefore, the answer is $3$.

Hence, the correct option is (B).

Note:As per in this question repeat gets to know that the question must be easier when the values are obtained and put on the graph whereas if we do this question only by calculation then there may or may not be any error from which I can suggest to always do this type of question by using graphical method.