F(x)= 3x^4 + 4x^3 - 12x^2 + pThe no. of real roots of f(x)=0 when p〈-32and no. of roots of f(x)=0 when 5〈p〈32.

To determine the number of real roots of the polynomial function \( F(x) = 3x^4 + 4x^3 - 12x^2 + p \) under the conditions \( p < -32 \) and \( 5 < p < 32 \), we can analyze the behavior of the function based on its derivative and the nature of polynomials.

Analyzing the Polynomial

The function \( F(x) \) is a quartic polynomial, which means it can have up to four real roots. The number of real roots can be influenced by the value of \( p \). To understand how \( p \) affects the roots, we first need to find the critical points of \( F(x) \) by taking its derivative.

Finding the Derivative

The derivative of \( F(x) \) is given by:

\( F'(x) = 12x^3 + 12x^2 - 24x \)

This can be factored as:

\( F'(x) = 12x(x^2 + x - 2) \)

Further factoring the quadratic gives:

\( F'(x) = 12x(x - 1)(x + 2) \)

Critical Points and Behavior

The critical points occur at \( x = 0 \), \( x = 1 \), and \( x = -2 \). These points divide the x-axis into intervals where we can analyze the sign of \( F'(x) \) to determine where \( F(x) \) is increasing or decreasing:

• For \( x < -2 \): \( F'(x) > 0 \) (increasing)
• For \( -2 < x < 0 \): \( F'(x) < 0 \) (decreasing)
• For \( 0 < x < 1 \): \( F'(x) > 0 \) (increasing)
• For \( x > 1 \): \( F'(x) > 0 \) (increasing)

This means that \( F(x) \) has a local maximum at \( x = -2 \) and a local minimum at \( x = 0 \). To find the values of \( F(x) \) at these critical points, we can substitute them back into the original function.

Evaluating Critical Points

Calculating \( F(-2) \) and \( F(0) \):

\( F(-2) = 3(-2)^4 + 4(-2)^3 - 12(-2)^2 + p = 48 - 32 - 48 + p = -32 + p \)

\( F(0) = p \)

Determining the Number of Real Roots

Now, let's analyze the conditions for the number of real roots based on the values of \( p \):

Case 1: When \( p < -32 \)

In this scenario:

• At \( x = -2 \), \( F(-2) = -32 + p < -32 - 32 = -64 \) (which is negative).
• At \( x = 0 \), \( F(0) = p < -32 \) (also negative).

Since \( F(x) \) is increasing for \( x < -2 \) and \( F(-2) < 0 \), it must cross the x-axis at least once before \( x = -2 \). As \( F(0) < 0 \) and \( F(x) \) is increasing for \( x > 0 \), it must cross the x-axis again after \( x = 0 \). Therefore, there are a total of **two real roots** when \( p < -32 \).

Case 2: When \( 5 < p < 32 \)

In this range:

• At \( x = -2 \), \( F(-2) = -32 + p > -32 + 5 = -27 \) (which is positive).
• At \( x = 0 \), \( F(0) = p > 5 \) (also positive).

Since \( F(-2) > 0 \) and \( F(0) > 0 \), and knowing that \( F(x) \) is increasing for \( x < -2 \), it must cross the x-axis at least once before \( x = -2 \). However, since both critical points yield positive values, there are no crossings after \( x = 0 \). Thus, there is **one real root** when \( 5 < p < 32 \).

Summary

To summarize:

• For \( p < -32 \), there are **two real roots**.
• For \( 5 < p < 32 \), there is **one real root**.

This analysis shows how the value of \( p \) influences the number of real roots of the polynomial function \( F(x) \). Understanding the behavior of polynomials through their derivatives and critical points is key to solving such problems.