Answer

Verified

456k+ views

**Hint:**This problem can be solved by using derivatives. Slope of the Tangent line of the curve will be obtained. Then based on the value of slope, angle of the tangent with the x-axis can be easily computed.

**Complete step-by-step answer:**

Equation of the curve given in the question is,

\[y = 2{x^7} + 3x + 5\]

Now, its first derivative will be done with respect to x, to get the value of $\dfrac{{dy}}{{dx}}$.

The given expression has three terms each with a simple exponential value of variable x or some constant.

So, $

\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(2{x^7} + 3x + 5) \\

\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(2{x^7}) + \dfrac{d}{{dx}}(3x) + \dfrac{d}{{dx}}(5)...(1) \\

\\

$

We know that,

$\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$

Thus by putting value of n as 7 we get,

$\dfrac{d}{{dx}}({x^7}) = 7{x^6}$

And similarly by putting value of n as 1 we get,

$\dfrac{d}{{dx}}({x^1}) = {x^0} = 1$

Also the derivative of the constant term is always 0.

Now, putting all above values in equation (1), we get

$

\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(2{x^7}) + \dfrac{d}{{dx}}(3x) + \dfrac{d}{{dx}}(5) \\

\Rightarrow \dfrac{{dy}}{{dx}} = 2 \times 7{x^6} + 3 \times 1 + 0 \\

\Rightarrow \dfrac{{dy}}{{dx}} = 14{x^6} + 3 \\

\\

$

The clear value of $\dfrac{{dy}}{{dx}}$ will be positive as, ${x^6}$ is always a positive for any real value of x.

Thus $\tan \theta $ will also be positive. So, $\theta $ means the angle of the tangent line with x-axis will lie in the first quadrant.

Thus, the tangent to the curve will definitely make an acute angle with the x-axis.

**So, the correct answer is “Option C”.**

**Note:**All trigonometric ratios are having specific properties of their values based on their quadrant. This property helps a lot to find the possible range of values of the angle by knowing the positive or negative signs of the trigonometric ratios. Also the first derivative of any curve will give the slope of the tangent line of the curve. So, in the above problem we have used these properties to get the correct solution.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

What percentage of the solar systems mass is found class 8 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE